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String Theory, An Introduction to the Bosonic String The two volumes that comprise...

Author:
Joseph Polchinski

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String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal ﬁeld theory. The ﬁrst four chapters introduce the central ideas of string theory, the tools of conformal ﬁeld theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactiﬁcation and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the ﬁniteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersymmetric string theories and goes on to a broad presentation of the important advances of recent years. The ﬁrst three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal ﬁeld theory. The ﬁnal four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries. An appendix summarizes the necessary background on fermions and supersymmetry. Both volumes contain an annotated reference section, emphasizing references that will be useful to the student, as well as a detailed glossary of important terms and concepts. Many exercises are included which are intended to reinforce the main points of the text and to bring in additional ideas. An essential text and reference for graduate students and researchers in theoretical physics, particle physics, and relativity with an interest in modern superstring theory. Joseph Polchinski received his Ph.D. from the University of California at Berkeley in 1980. After postdoctoral fellowships at the Stanford Linear Accelerator Center and Harvard, he joined the faculty at the University of Texas at Austin in 1984, moving to his present position of Professor of Physics at the University of California at Santa Barbara, and Permanent Member of the Institute for Theoretical Physics, in 1992. Professor Polchinski is not only a clear and pedagogical expositor, but is also a leading string theorist. His discovery of the importance of D-branes in 1995 is one of the most important recent contributions in this ﬁeld, and he has also made signiﬁcant contributions to many areas of quantum ﬁeld theory and to supersymmetric models of particle physics.

From reviews of the hardback editions: Volume 1 ‘. . . This is an impressive book. It is notable for its consistent line of development and the clarity and insight with which topics are treated . . . It is hard to think of a better text in an advanced graduate area, and it is rare to have one written by a master of the subject. It is worth pointing out that the book also contains a collection of useful problems, a glossary, and an unusually complete index.’ Physics Today ‘. . . the most comprehensive text addressing the discoveries of the superstring revolutions of the early to mid 1990s, which mark the beginnings of “modern” string theory.’ Donald Marolf, University of California, Santa Barbara, American Journal of Physics ‘Physicists believe that the best hope for a fundamental theory of nature – including unification of quantum mechanics with general relativity and elementary particle theory – lies in string theory. This elegant mathematical physics subject is expounded by Joseph Polchinski in two volumes from Cambridge University Press . . . Written for advanced students and researchers, this set provides thorough and up-to-date knowledge.’ American Scientist ‘We would like to stress the pedagogical value of the present book. The approach taken is modern and pleasantly systematic, and it covers a broad class of results in a unified language. A set of exercises at the end of each chapter complements the discussion in the main text. On the other hand, the introduction of techniques and concepts essential in the context of superstrings makes it a useful reference for researchers in the field.’ Mathematical Reviews ‘It amply fulfils the need to inspire future string theorists on their long slog and is destined to become a classic. It is a truly exciting enterprise and one hugely served by this magnificent book.’ David Bailin, The Times Higher Education Supplement Volume 2 ‘In summary, these volumes will provide . . . the standard text and reference for students and researchers in particle physics and relativity interested in the possible ramifications of modern superstring theory.’ Allen C. Hirshfeld, General Relativity and Gravitation ‘Polchinski is a major contributor to the exciting developments that have revolutionised our understanding of string theory during the past four years; he is also an exemplary teacher, as Steven Weinberg attests in his foreword. He has produced an outstanding two-volume text, with numerous exercises accompanying each chapter. It is destined to become a classic . . . magnificent.’ David Bailin, The Times Higher Education Supplement ‘The present volume succeeds in giving a detailed yet comprehensive account of our current knowledge of superstring dynamics. The topics covered range from the basic construction of the theories to the most recent discoveries on their non-perturbative behaviour. The discussion is remarkably self-contained (the volume even contains a useful appendix on spinors and supersymmetry in several dimensions), and thus may serve as an introduction to the subject, and as an excellent reference for researchers in the field.’ Mathematical Reviews

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General editors: P. V. Landshoff, D. R. Nelson, S. Weinberg S. J. Aarseth Gravitational N-Body Simulations J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach A. M. Anile Relativistic Fluids and Magneto-Fluids J. A. de Azc´arrage and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O. Babelon, D. Bernard and M. Talon Introduction to Classical Integrable Systems F. Bastianelli and P. van Nieuwenhuizen Path Integrals and Anomalies in Curved Space V. Belinkski and E. Verdaguer Gravitational Solitons J. Bernstein Kinetic Theory in the Expanding Universe G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space† M. Burgess Classical Covariant Fields S. Carlip Quantum Gravity in 2 + 1 Dimensions J. C. Collins Renormalization† M. Creutz Quarks, Gluons and Lattices† P. D. D’Eath Supersymmetric Quantum Cosmology F. de Felice and C. J. S. Clarke Relativity on Curved Manifolds† B. S. DeWitt Supermanifolds, 2nd edition† P. G. O. Freund Introduction to Supersymmetry† J. Fuchs Affine Lie Algebras and Quantum Groups† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y. Fujii and K. Maeda The Scalar – Tensor Theory of Gravitation A. S. Galperin, E. A. Ivanov, V. I. Orievetsky and E. S. Sokatchev Harmonic Superspace R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity† M. G¨ockeler and T. Sch¨ucker Differential Geometry, Gauge Theories and Gravity† C. G´omez, M. Ruiz Altaba and G. Sierra Quantum Groups in Two-Dimensional Physics M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 1: Introduction† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory, volume 2: Loop Amplitudes, Anomalies and Phenomenology† V. N. Gribov The Theory of Complex Angular Momenta S. W. Hawking and G. F. R. Ellis The Large-Scale Structure of Space-Time† F. Iachello and A. Arima The Interacting Boson Model F. Iachello and P. van Isacker The Interacting Boson–Fermion Model C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J.-M. Drouffe Statistical Field Theory, volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems† C. Johnson D-Branes J. I. Kapusta Finite-Temperature Field Theory† V. E. Korepin, A. G. Izergin and N. M. Boguliubov The Quantum Inverse Scattering Method and Correlation Functions† M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory N. Manton and P. Sutcliffe Topological Solitons N. H. March Liquid Metals: Concepts and Theory I. M. Montvay and G. M¨unster Quantum Fields on a Lattice† L. O’ Raifeartaigh Group Structure of Gauge Theories† T. Ortín Gravity and Strings A. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† R. Penrose and W. Rindler Spinors and Space-Time, volume 1: Two-Spinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and Space-Time, volume 2: Spinor and Twistor Methods in Space-Time Geometry† S. Pokorski Gauge Field Theories, 2nd edition J. Polchinski String Theory, volume 1: An Introduction to the Bosonic, String† J. Polchinski String Theory, volume 2: Superstring Theory and Beyond† V. N. Popov Functional Integrals and Collective Excitations† R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton† C. Rovelli Quantum Gravity W. C. Saslaw Gravitational Physics of Stellar and Galactic Systems† H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition J. M. Stewart Advanced General Relativity† A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells Jr Twistor Geometry and Field Theories† J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics †

Issued as a paperback

STRING THEORY VOLUME I An Introduction to the Bosonic String JOSEPH POLCHINSKI Institute for Theoretical Physics University of California at Santa Barbara

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521633031 © Cambridge University Press 2001, 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 1998 eBook (NetLibrary) ISBN-13 978-0-511-33821-2 ISBN-10 0-511-33821-X eBook (NetLibrary) ISBN-13 ISBN-10

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ISBN-13 ISBN-10

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to my parents

Contents

Foreword

xiii

Preface

xv

Notation

xviii

1 A ﬁrst look at strings 1.1 Why strings? 1.2 Action principles 1.3 The open string spectrum 1.4 Closed and unoriented strings Exercises

1 1 9 16 25 30

2 Conformal ﬁeld theory 2.1 Massless scalars in two dimensions 2.2 The operator product expansion 2.3 Ward identities and Noether’s theorem 2.4 Conformal invariance 2.5 Free CFTs 2.6 The Virasoro algebra 2.7 Mode expansions 2.8 Vertex operators 2.9 More on states and operators Exercises

32 32 36 41 43 49 52 58 63 68 74

3 3.1 3.2 3.3 3.4 3.5 3.6

77 77 82 84 90 97 101

The Polyakov path integral Sums over world-sheets The Polyakov path integral Gauge ﬁxing The Weyl anomaly Scattering amplitudes Vertex operators

ix

x

Contents

3.7 Strings in curved spacetime Exercises

108 118

4 The string spectrum 4.1 Old covariant quantization 4.2 BRST quantization 4.3 BRST quantization of the string 4.4 The no-ghost theorem Exercises

121 121 126 131 137 143

5 The string S-matrix 5.1 The circle and the torus 5.2 Moduli and Riemann surfaces 5.3 The measure for moduli 5.4 More about the measure Exercises

145 145 150 154 159 164

6 Tree-level amplitudes 6.1 Riemann surfaces 6.2 Scalar expectation values 6.3 The bc CFT 6.4 The Veneziano amplitude 6.5 Chan–Paton factors and gauge interactions 6.6 Closed string tree amplitudes 6.7 General results Exercises

166 166 169 176 178 184 192 198 204

7 One-loop amplitudes 7.1 Riemann surfaces 7.2 CFT on the torus 7.3 The torus amplitude 7.4 Open and unoriented one-loop graphs Exercises

206 206 208 216 222 229

8 Toroidal compactiﬁcation and T -duality 8.1 Toroidal compactiﬁcation in ﬁeld theory 8.2 Toroidal compactiﬁcation in CFT 8.3 Closed strings and T -duality 8.4 Compactiﬁcation of several dimensions 8.5 Orbifolds 8.6 Open strings 8.7 D-branes 8.8 T -duality of unoriented theories Exercises

231 231 235 241 249 256 263 268 277 280

Contents

xi

9 Higher order amplitudes 9.1 General tree-level amplitudes 9.2 Higher genus Riemann surfaces 9.3 Sewing and cutting world-sheets 9.4 Sewing and cutting CFTs 9.5 General amplitudes 9.6 String ﬁeld theory 9.7 Large order behavior 9.8 High energy and high temperature 9.9 Low dimensions and noncritical strings Exercises

283 283 290 294 300 305 310 315 317 322 327

Appendix A: A short course on path integrals A.1 Bosonic ﬁelds A.2 Fermionic ﬁelds Exercises

329 329 341 345

References

347

Glossary

359

Index

389

Outline of volume II

10

Type I and type II superstrings

11

The heterotic string

12

Superstring interactions

13

D-branes

14

Strings at strong coupling

15

Advanced CFT

16

Orbifolds

17

Calabi–Yau compactiﬁcation

18

Physics in four dimensions

19

Advanced topics

Appendix B: Spinors and SUSY in various dimensions

xii

Foreword

From the beginning it was clear that, despite its successes, the Standard Model of elementary particles would have to be embedded in a broader theory that would incorporate gravitation as well as the strong and electroweak interactions. There is at present only one plausible candidate for such a theory: it is the theory of strings, which started in the 1960s as a not-very-successful model of hadrons, and only later emerged as a possible theory of all forces. There is no one better equipped to introduce the reader to string theory than Joseph Polchinski. This is in part because he has played a signiﬁcant role in the development of this theory. To mention just one recent example: he discovered the possibility of a new sort of extended object, the ‘Dirichlet brane,’ which has been an essential ingredient in the exciting progress of the last few years in uncovering the relation between what had been thought to be diﬀerent string theories. Of equal importance, Polchinski has a rare talent for seeing what is of physical signiﬁcance in a complicated mathematical formalism, and explaining it to others. In looking over the proofs of this book, I was reminded of the many times while Polchinski was a member of the Theory Group of the University of Texas at Austin, when I had the beneﬁt of his patient, clear explanations of points that had puzzled me in string theory. I recommend this book to any physicist who wants to master this exciting subject. Steven Weinberg Series Editor Cambridge Monographs on Mathematical Physics 1998

xiii

Preface

When I ﬁrst decided to write a book on string theory, more than ten years ago, my memories of my student years were much more vivid than they are today. Still, I remember that one of the greatest pleasures was ﬁnding a text that made a diﬃcult subject accessible, and I hoped to provide the same for string theory. Thus, my ﬁrst purpose was to give a coherent introduction to string theory, based on the Polyakov path integral and conformal ﬁeld theory. No previous knowledge of string theory is assumed. I do assume that the reader is familiar with the central ideas of general relativity, such as metrics and curvature, and with the ideas of quantum ﬁeld theory through nonAbelian gauge symmetry. Originally a full course of quantum ﬁeld theory was assumed as a prerequisite, but it became clear that many students were eager to learn string theory as soon as possible, and that others had taken courses on quantum ﬁeld theory that did not emphasize the tools needed for string theory. I have therefore tried to give a self-contained introduction to those tools. A second purpose was to show how some of the simplest fourdimensional string theories connect with previous ideas for unifying the Standard Model, and to collect general results on the physics of fourdimensional string theories as derived from world-sheet and spacetime symmetries. New developments have led to a third goal, which is to introduce the recent discoveries concerning string duality, M-theory, D-branes, and black hole entropy. In writing a text such as this, there is a conﬂict between the need to be complete and the desire to get to the most interesting recent results as quickly as possible. I have tried to serve both ends. On the side of completeness, for example, the various path integrals in chapter 6 are calculated by three diﬀerent methods, and the critical dimension of the bosonic string is calculated in seven diﬀerent ways in the text and exercises. xv

xvi

Preface

On the side of eﬃciency, some shorter paths through these two volumes are suggested below. A particular issue is string perturbation theory. This machinery is necessarily a central subject of volume one, but it is somewhat secondary to the recent nonperturbative developments: the free string spectrum plus the spacetime symmetries are more crucial there. Fortunately, from string perturbation theory there is a natural route to the recent discoveries, by way of T -duality and D-branes. One possible course consists of chapters 1–3, section 4.1, chapters 5–8 (omitting sections 5.4 and 6.7), chapter 10, sections 11.1, 11.2, 11.6, 12.1, and 12.2, and chapters 13 and 14. This sequence, which I believe can be covered in two quarters, takes one from an introduction to string theory through string duality, M theory, and the simplest black hole entropy calculations. An additional shortcut is suggested at the end of section 5.1. Readers interested in T -duality and related stringy phenomena can proceed directly from section 4.1 to chapter 8. The introduction to Chan– Paton factors at the beginning of section 6.5 is needed to follow the discussion of the open string, and the one-loop vacuum amplitude, obtained in chapter 7, is needed to follow the calculation of the D-brane tension. Readers interested in supersymmetric strings can read much of chapters 10 and 11 after section 4.1. Again the introduction to Chan–Paton factors is needed to follow the open string discussion, and the one-loop vacuum amplitude is needed to follow the consistency conditions in sections 10.7, 10.8, and 11.2. Readers interested in conformal ﬁeld theory might read chapter 2, sections 6.1, 6.2, 6.7, 7.1, 7.2, 8.2, 8.3 (concentrating on the CFT aspects), 8.5, 10.1–10.4, 11.4, and 11.5, and chapter 15. Readers interested in four-dimensional string theories can follow most of chapters 16–19 after chapters 8, 10, and 11. In a subject as active as string theory — by one estimate the literature approaches 10 000 papers — there will necessarily be important subjects that are treated only brieﬂy, and others that are not treated at all. Some of these are represented by review articles in the lists of references at the end of each volume. The most important omission is probably a more complete treatment of compactiﬁcation on curved manifolds. Because the geometric methods of this subject are somewhat orthogonal to the quantum ﬁeld theory methods that are emphasized here, I have included only a summary of the most important results in chapters 17 and 19. Volume two of Green, Schwarz, and Witten (1987) includes a more extensive introduction, but this is a subject that has continued to grow in importance and clearly deserves an introductory book of its own. This work grew out of a course taught at the University of Texas

Preface

xvii

at Austin in 1987–8. The original plan was to spend a year turning the lecture notes into a book, but a desire to make the presentation clearer and more complete, and the distraction of research, got in the way. An early prospectus projected the completion date as June 1989 ± one month, oﬀ by 100 standard deviations. For eight years the expected date of completion remained approximately one year in the future, while one volume grew into two. Happily, ﬁnally, one of those deadlines didn’t slip. I have also used portions of this work in a course at the University of California at Santa Barbara, and at the 1994 Les Houches, 1995 Trieste, and 1996 TASI schools. Portions have been used for courses by Nathan Seiberg and Michael Douglas (Rutgers), Steven Weinberg (Texas), Andrew ˜ Paulo) Strominger and Juan Maldacena (Harvard), Nathan Berkovits (Sao and Martin Einhorn (Michigan). I would like to thank those colleagues and their students for very useful feedback. I would also like to thank Steven Weinberg for his advice and encouragement at the beginning of this project, Shyamoli Chaudhuri for a thorough reading of the entire manuscript, and to acknowledge the support of the Departments of Physics at UT Austin and UC Santa Barbara, the Institute for Theoretical Physics at UC Santa Barbara, and the National Science Foundation. During the extended writing of this book, dozens of colleagues have helped to clarify my understanding of the subjects covered, and dozens of students have suggested corrections and other improvements. I began to try to list the members of each group and found that it was impossible. Rather than present a lengthy but incomplete list here, I will keep an updated list at the erratum website http://www.itp.ucsb.edu/˜joep/bigbook.html. In addition, I would like to thank collectively all who have contributed to the development of string theory; volume two in particular seems to me to be largely a collection of beautiful results derived by many physicists. String theory (and the entire base of physics upon which it has been built) is one of mankind’s great achievements, and it has been my privilege to try to capture its current state. Finally, to complete a project of this magnitude has meant many sacriﬁces, and these have been shared by my family. I would like to thank Dorothy, Steven, and Daniel for their understanding, patience, and support. Joseph Polchinski Santa Barbara, California 1998

Notation

This book uses the +++ conventions of Misner, Thorne, and Wheeler (1973). In particular, the signature of the metric is (− + + . . . +). The constants ¯h and c are set to 1, but the Regge slope α is kept explicit. A bar ¯ is used to denote the conjugates of world-sheet coordinates and moduli (such as z, τ, and q), but a star ∗ is used for longer expressions. A bar on a spacetime fermion ﬁeld is the Dirac adjoint (this appears only in volume two), and a bar on a world-sheet operator is the Euclidean adjoint (deﬁned in section 6.7). For the degrees of freedom on the string, the following terms are treated as synonymous: holomorphic = left-moving, antiholomorphic = right-moving, as explained in section 2.1. Our convention is that the supersymmetric side of the heterotic string is right-moving. Antiholomorphic operators are designated by tildes ˜; as explained in section 2.3, these are not the adjoints of holomorphic operators. Note also the following conventions: d2 z ≡ 2dxdy ,

1 δ 2 (z, ¯z ) ≡ δ(x)δ(y) , 2

where z = x + iy is any complex variable; these diﬀer from most of the literature, where the coeﬃcient is 1 in each deﬁnition. Spacetime actions are written as S and world-sheet actions as S . This presents a problem for D-branes, which are T -dual to the former and S -dual to the latter; S has been used arbitrarily. The spacetime metric is Gµν , while the world-sheet metric is γab (Minkowskian) or gab (Euclidean). In volume one, the spacetime Ricci tensor is Rµν and the world-sheet Ricci tensor is Rab . In volume two the former appears often and the latter never, so we have changed to Rµν for the spacetime Ricci tensor. xviii

Notation

xix

The following are used: ≡ ∼ = ≈ ∼

deﬁned as equivalent to approximately equal to equal up to nonsingular terms (in OPEs), or rough correspondence.

1 A ﬁrst look at strings

1.1

Why strings?

One of the main themes in the history of science has been uniﬁcation. Time and again diverse phenomena have been understood in terms of a small number of underlying principles and building blocks. The principle that underlies our current understanding of nature is quantum ﬁeld theory, quantum mechanics with the basic observables living at spacetime points. In the late 1940s it was shown that quantum ﬁeld theory is the correct framework for the uniﬁcation of quantum mechanics and electromagnetism. By the early 1970s it was understood that the weak and strong nuclear forces are also described by quantum ﬁeld theory. The full theory, the S U(3) × S U(2) × U(1) Model or Standard Model, has been conﬁrmed repeatedly in the ensuing years. Combined with general relativity, this theory is consistent with virtually all physics down to the scales probed by particle accelerators, roughly 10−16 cm. It also passes a variety of indirect tests that probe to shorter distances, including precision tests of quantum electrodynamics, searches for rare meson decays, limits on neutrino masses, limits on axions (light weakly interacting particles) from astrophysics, searches for proton decay, and gravitational limits on the couplings of massless scalars. In each of these indirect tests new physics might well have appeared, but in no case has clear evidence for it yet been seen; at the time of writing, the strongest sign is the solar neutrino problem, suggesting nonzero neutrino masses. The Standard Model (plus gravity) has a fairly simple structure. There are four interactions based on local invariance principles. One of these, gravitation, is mediated by the spin-2 graviton, while the other three are mediated by the spin-1 S U(3) × S U(2) × U(1) gauge bosons. In addition, the theory includes the spin-0 Higgs boson needed for symmetry breaking, and the quarks and leptons, ﬁfteen multiplets of spin- 12 fermions in three 1

2

1 A ﬁrst look at strings

generations of ﬁve. The dynamics is governed by a Lagrangian that depends upon roughly twenty free parameters such as the gauge and Yukawa couplings. In spite of its impressive successes, this theory is surely not complete. First, it is too arbitrary: why does this particular pattern of gauge ﬁelds and multiplets exist, and what determines the parameters in the Lagrangian? Second, the union of gravity with quantum theory yields a nonrenormalizable quantum ﬁeld theory, a strong signal that new physics should appear at very high energy. Third, even at the classical level the theory breaks down at the singularities of general relativity. Fourth, the theory is in a certain sense unnatural: some of the parameters in the Lagrangian are much smaller than one would expect them to be. It is these problems, rather than any positive experimental evidence, that presently must guide us in our attempts to ﬁnd a more complete theory. One seeks a principle that uniﬁes the ﬁelds of the Standard Model in a simpler structure, and resolves the divergence and naturalness problems. Several promising ideas have been put forward. One is grand uniﬁcation. This combines the three gauge interactions into one and the ﬁve multiplets of each generation into two or even one. It also successfully predicts one of the free parameters (the weak mixing angle) and possibly another (the bottom-tau mass ratio). A second idea is that spacetime has more than four dimensions, with the additional ones so highly curved as to be undetectable at current energies. This is certainly a logical possibility, since spacetime geometry is dynamical in general relativity. What makes it attractive is that a single higher-dimensional ﬁeld can give rise to many four-dimensional ﬁelds, diﬀering in their polarization (which can point along the small dimensions or the large) and in their dependence on the small dimensions. This opens the possibility of unifying the gauge interactions and gravity (the Kaluza–Klein mechanism). It also gives a natural mechanism for producing generations, repeated copies of the same fermion multiplets. A third unifying principle is supersymmetry, which relates ﬁelds of diﬀerent spins and statistics, and which helps with the divergence and naturalness problems. Each of these ideas — grand uniﬁcation, extra dimensions, and supersymmetry — has attractive features and is consistent with the various tests of the Standard Model. It is plausible that these will be found as elements of a more complete theory of fundamental physics. It is clear, however, that something is still missing. Applying these ideas, either singly or together, has not led to theories that are substantially simpler or less arbitrary than the Standard Model. Short-distance divergences have been an important issue many times in quantum ﬁeld theory. For example, they were a key clue leading from the Fermi theory of the weak interaction to the Weinberg–Salam theory. Let

3

1.1 Why strings?

(a)

(b)

(c)

Fig. 1.1. (a) Two particles propagating freely. (b) Correction from one-graviton exchange. (c) Correction from two-graviton exchange.

us look at the short-distance problem of quantum gravity, which can be understood from a little dimensional analysis. Figure 1.1 shows a process, two particles propagating, and corrections due to one-graviton exchange and two-graviton exchange. The one-graviton exchange is proportional to Newton’s constant GN . The ratio of the one-graviton correction to the original amplitude must be governed by the dimensionless combination GN E 2 ¯h−1 c−5 , where E is the characteristic energy of the process; this is the only dimensionless combination that can be formed from the parameters in the problem. Throughout this book we will use units in which h¯ = c = 1, deﬁning the Planck mass −1/2

MP = GN

= 1.22 × 1019 GeV

(1.1.1)

and the Planck length MP−1 = 1.6 × 10−33 cm .

(1.1.2)

The ratio of the one-graviton to the zero-graviton amplitude is then of order (E/MP )2 . From this dimensional analysis one learns immediately that the quantum gravitational correction is an irrelevant interaction, meaning that it grows weaker at low energy, and in particular is negligible at particle physics energies of hundreds of GeV. By the same token, the coupling grows stronger at high energy and at E > MP perturbation theory breaks down. In the two-graviton correction of ﬁgure 1.1(c) there is a sum over intermediate states. For intermediate states of high energy E , the ratio of the two-graviton to the zero-graviton amplitude is on dimensional grounds of order G2N E 2

dE E =

E2 MP4

dE E ,

(1.1.3)

4

1 A ﬁrst look at strings

which diverges if the theory is extrapolated to arbitrarily high energies. In position space this divergence comes from the limit where all the graviton vertices become coincident. The divergence grows worse with each additional graviton — this is the problem of nonrenormalizability. There are two possible resolutions. The ﬁrst is that the divergence is due to expanding in powers of the interaction and disappears when the theory is treated exactly. In the language of the renormalization group, this would be a nontrivial UV ﬁxed point. The second is that the extrapolation of the theory to arbitrarily high energies is incorrect, and beyond some energy the theory is modiﬁed in a way that smears out the interaction in spacetime and softens the divergence. It is not known whether quantum gravity has a nontrivial UV ﬁxed point, but there are a number of reasons for concentrating on the second possibility. One is history — the same kind of divergence problem in the Fermi theory of the weak interaction was a sign of new physics, the contact interaction between the fermions resolving at shorter distance into the exchange of a gauge boson. Another is that we need a more complete theory in any case to account for the patterns in the Standard Model, and it is reasonable to hope that the same new physics will solve the divergence problem of quantum gravity. In quantum ﬁeld theory it is not easy to smear out interactions in a way that preserves the consistency of the theory. We know that Lorentz invariance holds to very good approximation, and this means that if we spread the interaction in space we spread it in time as well, with consequent loss of causality or unitarity. Moreover we know that Lorentz invariance is actually embedded in a local symmetry, general coordinate invariance, and this makes it even harder to spread the interaction out without producing inconsistencies. In fact, there is presently only one way known to spread out the gravitational interaction and cut oﬀ the divergence without spoiling the consistency of the theory. This is string theory, illustrated in ﬁgure 1.2. In this theory the graviton and all other elementary particles are onedimensional objects, strings, rather than points as in quantum ﬁeld theory. Why this should work and not anything else is not at all obvious a priori, but as we develop the theory we will see how it comes about.1 Perhaps we merely suﬀer from a lack of imagination, and there are many other consistent theories of gravity with a short-distance cutoﬀ. However, experience has shown that divergence problems in quantum ﬁeld theory 1

There is an intuitive answer to at least one common question: why not membranes, two- or higher-dimensional objects? The answer is that as we spread out particles in more dimensions we reduce the spacetime divergences, but encounter new divergences coming from the increased number of internal degrees of freedom. One dimension appears to be the unique case where both the spacetime and internal divergences are under control. However, as we will discuss in chapter 14, the membrane idea has resurfaced in somewhat transmuted form as matrix theory.

1.1 Why strings?

5

(a)

(b)

(c)

Fig. 1.2. (a) Closed string. (b) Open string. (c) The loop amplitude of ﬁg. 1.1(c) in string theory. Each particle world-line becomes a cylinder, and the interactions no longer occur at points. (The cross-sections on the intermediate lines are included only for perspective.)

are not easily resolved, so if we have even one solution we should take it very seriously. Indeed, we are fortunate that consistency turns out to be such a restrictive principle, since the uniﬁcation of gravity with the other interactions takes place at such high energy, MP , that experimental tests will be diﬃcult and indirect. So what do we ﬁnd if we pursue this idea? In a word, the result is remarkable. String theory dovetails beautifully with the previous ideas for explaining the patterns in the Standard Model, and does so with a structure more elegant and uniﬁed than in quantum ﬁeld theory. In particular, if one tries to construct a consistent relativistic quantum theory of one-dimensional objects one ﬁnds: 1. Gravity. Every consistent string theory must contain a massless spin-2 state, whose interactions reduce at low energy to general relativity. 2. A consistent theory of quantum gravity, at least in perturbation theory. As we have noted, this is in contrast to all known quantum ﬁeld theories of gravity. 3. Grand uniﬁcation. String theories lead to gauge groups large enough to include the Standard Model. Some of the simplest string theories lead to the same gauge groups and fermion representations that arise in the uniﬁcation of the Standard Model.

6

1 A ﬁrst look at strings 4. Extra dimensions. String theory requires a deﬁnite number of spacetime dimensions, ten.2 The ﬁeld equations have solutions with four large ﬂat and six small curved dimensions, with four-dimensional physics that resembles the Standard Model. 5. Supersymmetry. Consistent string theories require spacetime supersymmetry, as either a manifest or a spontaneously broken symmetry. 6. Chiral gauge couplings. The gauge interactions in nature are parity asymmetric (chiral). This has been a stumbling block for a number of previous unifying ideas: they required parity symmetric gauge couplings. String theory allows chiral gauge couplings. 7. No free parameters. String theory has no adjustable constants. 8. Uniqueness. Not only are there no continuous parameters, but there is no discrete freedom analogous to the choice of gauge group and representations in ﬁeld theory: there is a unique string theory.

In addition one ﬁnds a number of other features, such as an axion, and hidden gauge groups, that have played a role in ideas for uniﬁcation. This is a remarkable list, springing from the simple supposition of onedimensional objects. The ﬁrst two points alone would be of great interest. The next four points come strikingly close to the picture one arrives at in trying to unify the Standard Model. And as indicated by the last two points, string theory accomplishes this with a structure that is tighter and less arbitrary than in quantum ﬁeld theory, supplying the element missing in the previous ideas. The ﬁrst point is a further example of this tightness: string theory must have a graviton, whereas in ﬁeld theory this and other ﬁelds are combined in a mix-and-match fashion. String theory further has connections to many areas of mathematics, and has led to the discovery of new and unexpected relations among them. It has rich connections to the recent discoveries in supersymmetric quantum ﬁeld theory. String theory has also begun to address some of the deeper questions of quantum gravity, in particular the quantum mechanics of black holes. Of course, much remains to be done. String theory may resemble the real world in its broad outlines, but a decisive test still seems to be far away. The main problem is that while there is a unique theory, it has an enormous number of classical solutions, even if we restrict attention 2

To be precise, string theory modiﬁes the notions of spacetime topology and geometry, so what we mean by a dimension here is generalized. Also, we will see that ten dimensions is the appropriate number for weakly coupled string theory, but that the picture can change at strong coupling.

1.1 Why strings?

7

to solutions with four large ﬂat dimensions. Upon quantization, each of these is a possible ground state (vacuum) for the theory, and the fourdimensional physics is diﬀerent in each. It is known that quantum eﬀects greatly reduce the number of stable solutions, but a full understanding of the dynamics is not yet in hand. It is worth recalling that even in the Standard Model, the dynamics of the vacuum plays an important role in the physics we see. In the electroweak interaction, the fact that the vacuum is less symmetric than the Hamiltonian (spontaneous symmetry breaking) plays a central role. In the strong interaction, large ﬂuctuating gauge ﬁelds in the vacuum are responsible for quark conﬁnement. These phenomena in quantum ﬁeld theory arise from having a quantum system with many degrees of freedom. In string theory there are seemingly many more degrees of freedom, and so we should expect even richer dynamics. Beyond this, there is the question, ‘what is string theory?’ Until recently our understanding of string theory was limited to perturbation theory, small numbers of strings interacting weakly. It was not known how even to deﬁne the theory at strong coupling. There has been a suspicion that the degrees of freedom that we use at weak coupling, one-dimensional objects, are not ultimately the simplest or most complete way to understand the theory. In the past few years there has been a great deal of progress on these issues, growing largely out of the systematic application of the constraints imposed by supersymmetry. We certainly do not have a complete understanding of the dynamics of strongly coupled strings, but it has become possible to map out in detail the space of vacua (when there is enough unbroken supersymmetry) and this has led to many surprises. One is the absolute uniqueness of the theory: whereas there are several weakly coupled string theories, all turn out to be limits in the space of vacua of a single theory. Another is a limit in which spacetime becomes elevendimensional, an interesting number from the point of view of supergravity but impossible in weakly coupled string theory. It has also been understood that the theory contains new extended objects, D-branes, and this has led to the new understanding of black hole quantum mechanics. All this also gives new and unexpected clues as to the ultimate nature of the theory. In summary, we are fortunate that so many approaches seem to converge on a single compelling idea. Whether one starts with the divergence problem of quantum gravity, with attempts to account for the patterns in the Standard Model, or with a search for new symmetries or mathematical structures that may be useful in constructing a uniﬁed theory, one is led to string theory.

8

1 A ﬁrst look at strings Outline

The goal of these two volumes is to provide a complete introduction to string theory, starting at the beginning and proceeding through the compactiﬁcation to four dimensions and to the latest developments in strongly coupled strings. Volume one is an introduction to bosonic string theory. This is not a realistic theory — it does not have fermions, and as far as is known has no stable ground state. The philosophy here is the same as in starting a course on quantum ﬁeld theory with a thorough study of scalar ﬁeld theory. That is also not the theory one is ultimately interested in, but it provides a simple example for developing the unique dynamical and technical features of quantum ﬁeld theory before introducing the complications of spin and gauge invariance. Similarly, a thorough study of bosonic string theory will give us a framework to which we can in short order add the additional complications of fermions and supersymmetry. The rest of chapter 1 is introductory. We present ﬁrst the action principle for the dynamics of string. We then carry out a quick and heuristic quantization using light-cone gauge, to show the reader some of the important aspects of the string spectrum. Chapters 2–7 are the basic introduction to bosonic string theory. Chapter 2 introduces the needed technical tools in the world-sheet quantum ﬁeld theory, such as conformal invariance, the operator product expansion, and vertex operators. Chapters 3 and 4 carry out the covariant quantization of the string, starting from the Polyakov path integral. Chapters 5–7 treat interactions, presenting the general formalism and applying it to tree-level and one-loop amplitudes. Chapter 8 treats the simplest compactiﬁcation of string theory, making some of the dimensions periodic. In addition to the phenomena that arise in compactiﬁed ﬁeld theory, such as Kaluza–Klein gauge symmetry, there is also a great deal of ‘stringy’ physics, including enhanced gauge symmetries, T -duality, and D-branes. Chapter 9 treats higher order amplitudes. The ﬁrst half outlines the argument that string theory in perturbation theory is ﬁnite and unitary as advertised; the second half presents brief treatments of a number of advanced topics, such as string ﬁeld theory. Appendix A is an introduction to path integration, so that our use of quantum ﬁeld theory is self-contained. Volume two treats supersymmetric string theories, focusing ﬁrst on ten-dimensional and other highly symmetric vacua, and then on realistic four-dimensional vacua. In chapters 10–12 we extend the earlier introduction to the supersymmetric string theories, developing the type I, II, and heterotic superstrings and their interactions. We then introduce the latest results in these subjects. Chapter 13 develops the properties and dynamics of D-branes, still

1.2 Action principles

9

using the tools of string perturbation theory as developed earlier in the book. Chapter 14 then uses arguments based on supersymmetry to understand strongly coupled strings. We ﬁnd that the strongly coupled limit of any string theory is described by a dual weakly coupled string theory, or by a new eleven-dimensional theory known provisionally as M-theory. We discuss the status of the search for a complete formulation of string theory and present one promising idea, m(atrix) theory. We brieﬂy discuss the quantum mechanics of black holes, carrying out the simplest entropy calculation. Chapter 15 collects a number of advanced applications of the various world-sheet symmetry algebras. Chapters 16 and 17 present four-dimensional string theories based on orbifold and Calabi–Yau compactiﬁcations. The goal is not an exhaustive treatment but rather to make contact between the simplest examples and the uniﬁcation of the Standard Model. Chapter 18 collects results that hold in wide classes of string theories, using general arguments based on worldsheet and spacetime gauge symmetries. Chapter 19 consists of advanced topics, including (2,2) world-sheet supersymmetry, mirror symmetry, the conifold transition, and the strong-coupling behavior of some compactiﬁed theories. Annotated reference lists appear at the end of each volume. I have tried to assemble a selection of articles, particularly reviews, that may be useful to the student. A glossary also appears at the end of each volume. 1.2

Action principles

We want to study the classical and quantum mechanics of a one-dimensional object, a string. The string moves in D ﬂat spacetime dimensions, with metric ηµν = diag(−, +, +, · · · , +). It is useful to review ﬁrst the classical mechanics of a zero-dimensional object, a relativistic point particle. We can describe the motion of a particle by giving its position in terms of D−1 functions of time, X(X 0 ). This hides the covariance of the theory though, so it is better to introduce a parameter τ along the particle’s world-line and describe the motion in spacetime by D functions X µ (τ). The parameterization is arbitrary: a diﬀerent parameterization of the same path is physically equivalent, and all physical quantities must be independent of this choice. That is, for any monotonic function τ (τ), the two paths X µ and X µ are the same, where X µ (τ (τ)) = X µ (τ) .

(1.2.1)

We are trading a less symmetric description for a more symmetric but redundant one, which is often a useful step. Figure 1.3(a) shows a parameterized world-line.

10

1 A ﬁrst look at strings

X0

X0

τ =2

τ =2 τ =1

τ =1 τ= 0 τ=−1 τ =−2

τ= 0 τ=−1 τ =−2

σ =0

σ =l

(b)

(a)

X

X

Fig. 1.3. (a) Parameterized world-line of a point particle. (b) Parameterized world-sheet of an open string.

The simplest Poincar´e-invariant action that does not depend on the parameterization would be proportional to the proper time along the world-line, Spp = −m

˙ µX ˙ µ )1/2 , dτ (−X

(1.2.2)

where a dot denotes a τ-derivative. The variation of the action, after an integration by parts, is δSpp = −m

dτ ˙uµ δX µ ,

(1.2.3)

where ˙ ν )−1/2 ˙ µ (−X ˙ νX uµ = X

(1.2.4)

is the normalized D-velocity. The equation of motion ˙uµ = 0 thus describes free motion. The normalization constant m is the particle’s mass, as can be checked by looking at the nonrelativistic limit (exercise 1.1). The action can be put in another useful form by introducing an additional ﬁeld on the world-line, an independent world-line metric γττ (τ). It will be convenient to work with the tetrad η(τ) = (−γττ (τ))1/2 , which is deﬁned to be positive. We use the general relativity term tetrad in any number of dimensions, even though its root means ‘four.’ Then

1 ˙ µX ˙ µ − ηm2 . dτ η −1 X (1.2.5) 2 This action has the same symmetries as the earlier action Spp , namely Poincar´e invariance and world-line reparameterization invariance. Under Spp =

1.2 Action principles

11

the latter, η(τ) transforms as η (τ )dτ = η(τ)dτ .

(1.2.6)

The equation of motion from varying the tetrad is ˙ µ /m2 . ˙ µX η 2 = −X

(1.2.7)

becomes the earlier S . InciUsing this to eliminate η(τ), the action Spp pp dentally, Spp makes perfect sense for massless particles, while Spp does not work in that case. are equivalent classically, it is hard to Although the actions Spp and Spp make sense of Spp in a path integral because of its complicated form, with is quadratic in derivatives inside the square root. On the other hand Spp derivatives and its path integral is fairly easily evaluated, as we will see in later chapters. Presumably, any attempt to deﬁne the quantum theory for path integral, and we will Spp will lead to a result equivalent to the Spp take the latter as our starting point in deﬁning the quantum theory. A one-dimensional object will sweep out a two-dimensional worldsheet, which can be described in terms of two parameters, X µ (τ, σ) as in ﬁgure 1.3(b). As in the case of the particle, we insist that physical quantities such as the action depend only on the embedding in spacetime and not on the parameterization. Not only is it attractive that the theory should have this property, but we will see that it is necessary in a consistent relativistic quantum theory. The simplest invariant action, the Nambu–Goto action, is proportional to the area of the world-sheet. In order to express this action in terms of X µ (τ, σ), deﬁne ﬁrst the induced metric hab where indices a, b, . . . run over values (τ, σ):

hab = ∂a X µ ∂b Xµ .

(1.2.8)

Then the Nambu–Goto action is

SNG =

M

dτdσ LNG ,

(1.2.9a)

1 (− det hab )1/2 , (1.2.9b) 2πα where M denotes the world-sheet. The constant α , which has units of spacetime-length-squared, is the Regge slope, whose signiﬁcance will be seen later. The tension T of the string is related to the Regge slope by LNG = −

T =

1 2πα

(1.2.10)

(exercise 1.1). Let us now consider the symmetries of the action, transformations of X µ (τ, σ) such that SNG [X ] = SNG [X]. These are:

12

1 A ﬁrst look at strings 1. The isometry group of ﬂat spacetime, the D-dimensional Poincar´e group, X µ (τ, σ) = Λµν X ν (τ, σ) + aµ

(1.2.11)

with Λµν a Lorentz transformation and aµ a translation. 2. Two-dimensional coordinate invariance, often called diﬀeomorphism (diﬀ) invariance. For new coordinates (τ (τ, σ), σ (τ, σ)), the transformation is X µ (τ , σ ) = X µ (τ, σ) .

(1.2.12)

The Nambu–Goto action is analogous to the point particle action Spp , with derivatives in the square root. Again we can simplify it by introducing an independent world-sheet metric γab (τ, σ). Henceforth ‘metric’ will always mean γab (unless we specify ‘induced’), and indices will always be raised and lowered with this metric. We will take γab to have Lorentzian signature (−, +). The action is

1 dτ dσ (−γ)1/2 γ ab ∂a X µ ∂b Xµ , (1.2.13) 4πα M where γ = det γab . This is the Brink–Di Vecchia–Howe–Deser–Zumino action, or Polyakov action for short. It was found by Brink, Di Vecchia, and Howe and by Deser and Zumino, in the course of deriving a generalization with local world-sheet supersymmetry. Its virtues, especially for path integral quantization, were emphasized by Polyakov. To see the equivalence to SNG , use the equation of motion obtained by varying the metric, SP [X, γ] = −

1 1/2 ab cd 1 dτ dσ (−γ) δγ h − γ γ h (1.2.14) ab cd , 2 ab 4πα M where hab is again the induced metric (1.2.8). We have used the general relation for the variation of a determinant,

δγ SP [X, γ] = −

δγ = γγ ab δγab = −γγab δγ ab .

(1.2.15)

hab = 12 γab γ cd hcd .

(1.2.16)

Then δγ SP = 0 implies

Dividing this equation by the square root of minus its determinant gives hab (−h)−1/2 = γab (−γ)−1/2 ,

(1.2.17)

so that γab is proportional to the induced metric. This in turn can be used to eliminate γab from the action, SP [X, γ] → −

1 2πα

dτ dσ (−h)1/2 = SNG [X] .

(1.2.18)

1.2 Action principles

13

The action SP has the following symmetries: 1. D-dimensional Poincar´e invariance: X (τ, σ) = Λµν X ν (τ, σ) + aµ , γab (τ, σ) = γab (τ, σ) . µ

(1.2.19)

2. Diﬀ invariance: X (τ , σ ) = X µ (τ, σ) , µ

∂σ c ∂σ d γ (τ , σ ) = γab (τ, σ) , ∂σ a ∂σ b cd for new coordinates σ a (τ, σ).

(1.2.20)

3. Two-dimensional Weyl invariance: X (τ, σ) = X µ (τ, σ) , γab (τ, σ) = exp(2ω(τ, σ)) γab (τ, σ) , µ

(1.2.21)

for arbitrary ω(τ, σ) . The Weyl invariance, a local rescaling of the world-sheet metric, has no analog in the Nambu–Goto form. We can understand its appearance by looking back at the equation of motion (1.2.17) used to relate the Polyakov and Nambu–Goto actions. This does not determine γab completely but only up to a local rescaling, so Weyl-equivalent metrics correspond to the same embedding in spacetime. This is an extra redundancy of the Polyakov formulation, in addition to the obvious diﬀ invariance. The variation of the action with respect to the metric deﬁnes the energymomentum tensor, T ab (τ, σ) = −4π(−γ)−1/2

δ SP δγab

1 = − (∂a X µ ∂b Xµ − 12 γ ab ∂c X µ ∂c Xµ ) . (1.2.22) α This has an extra factor of −2π relative to the usual deﬁnition in ﬁeld theory; this is a standard and convenient convention in string theory. It is conserved, ∇a T ab = 0, as a consequence of diﬀ invariance. The invariance of SP under arbitrary Weyl transformations further implies that γab

δ SP = 0 ⇒ Taa = 0 . δγab

(1.2.23)

The actions SNG and SP deﬁne two-dimensional ﬁeld theories on the string world-sheet. In string theory, we will see that amplitudes for spacetime processes are given by matrix elements in the two-dimensional quantum ﬁeld theory on the world-sheet. While our interest is four-dimensional

14

1 A ﬁrst look at strings

spacetime, most of the machinery we use in string perturbation theory is two-dimensional. From the point of view of the world-sheet, the coordinate transformation law (1.2.12) deﬁnes X µ (τ, σ) as a scalar ﬁeld, with µ going along for the ride as an internal index. From the two-dimensional point of view the Polyakov action describes massless Klein–Gordon scalars X µ covariantly coupled to the metric γab . Also from this point of view, the Poincar´e invariance is an internal symmetry, meaning that it acts on ﬁelds at ﬁxed τ, σ. Varying γab in the action gives the equation of motion Tab = 0 .

(1.2.24)

Varying X µ gives the equation of motion ∂a [(−γ)1/2 γ ab ∂b X µ ] = (−γ)1/2 ∇2 X µ = 0 .

(1.2.25)

For world-sheets with boundary there is also a surface term in the variation of the action. To be speciﬁc, take the coordinate region to be −∞<τ<∞ ,

0≤σ≤.

(1.2.26)

We will think of τ as a time variable and σ as spatial, so this is a single string propagating without sources. Then 1 δSP = 2πα

∞

−∞

dτ 0

dσ (−γ)1/2 δX µ ∇2 Xµ

σ= ∞ 1 1/2 µ σ − dτ (−γ) δX ∂ X . (1.2.27) µ σ=0 2πα −∞ In the interior, the equation of motion is as above. The boundary term vanishes if

∂σ X µ (τ, 0) = ∂σ X µ (τ, ) = 0 . These are Neumann boundary conditions on na ∂a X µ = 0

X µ.

(1.2.28)

Stated more covariantly,

on ∂M ,

(1.2.29)

where na is normal to the boundary ∂M. The ends of the open string move freely in spacetime. The surface term in the equation of motion will also vanish if we impose X µ (τ, ) = X µ (τ, 0) , γab (τ, ) = γab (τ, 0) .

∂σ X µ (τ, ) = ∂σ X µ (τ, 0) ,

(1.2.30a) (1.2.30b)

That is, the ﬁelds are periodic. There is no boundary; the endpoints are joined to form a closed loop. The open string boundary condition (1.2.28) and closed string boundary condition (1.2.30) are the only possibilities consistent with D-dimensional

15

1.2 Action principles

Poincar´e invariance and the equations of motion. If we relax the condition of Poincar´e invariance then there are other possibilities which will be important later. Some of these are explored in the exercises. The Nambu–Goto and Polyakov actions may be the simplest with the given symmetries, but simplicity of the action is not the right criterion in quantum ﬁeld theory. Symmetry, not simplicity, is the key idea: for both physical and technical reasons we must usually consider the most general local action consistent with all of the symmetries of the theory. Let us try to generalize the Polyakov action, requiring that the symmetries be maintained and that the action be polynomial in derivatives (else it is hard to make sense of the quantum theory).3 Global Weyl invariance, ω(τ, σ) = constant, requires that the action have one more factor of γ ab than γab to cancel the variation of (−γ)1/2 . The extra upper indices can only contract with derivatives, so each term has precisely two derivatives. The coordinate invariance and Poincar´e invariance then allow one additional term beyond the original SP :

1 χ= dτ dσ (−γ)1/2 R , (1.2.31) 4π M where R is the Ricci scalar constructed from γab . Under a local Weyl rescaling, (−γ )1/2 R = (−γ)1/2 (R − 2∇2 ω) .

(1.2.32)

The variation is a total derivative, because (−γ)1/2 ∇a v a = ∂a ((−γ)1/2 v a ) for any v a . The integral (1.2.31) is therefore invariant for a world-sheet without boundary. With boundaries an additional surface term is needed (exercise 1.3). Since χ is allowed by the symmetries we will include it in the action: SP = SP − λχ . =−

dτ dσ (−γ) M

1/2

1 ab λ γ ∂a X µ ∂b X µ + R 4πα 4π

.

(1.2.33)

This is the most general (diﬀ×Weyl)-invariant and Poincar´e-invariant action with these ﬁelds and symmetries. For the moment the discussion of the symmetries is classical — we are ignoring possible quantum anomalies, which will be considered in chapter 3. The action SP looks like the Hilbert action for the metric, (−γ)1/2 R, minimally coupled to D massless scalar ﬁelds X µ . However, in two dimen3

If we relax this requirement then there are various additional possibilities, such as invariants constructed from the curvature of the induced metric hab . These would be expected to appear for such one-dimensional objects as electric and magnetic ﬂux tubes, and in that case the path integral makes sense because the thickness of the object provides a cutoﬀ. But for inﬁnitely thin strings, which is what we have in fundamental string theory, this is not possible.

16

1 A ﬁrst look at strings

sions, the Hilbert action depends only on the topology of the world-sheet and does not give dynamics to the metric. To see this, we use the result from general relativity that its variation is proportional to Rab − 12 γab R. In two dimensions, the symmetries of the curvature tensor imply that Rab = 12 γab R and so this vanishes: the Hilbert action is invariant under any continuous change in the metric. It does have some global signiﬁcance, as we will see in the chapter 3. There are various further ways one might try to generalize the theory. One, suggested by our eventual interest in D = 4, is to allow more than four X µ ﬁelds but require only four-dimensional Poincar´e invariance. A second, motivated by the idea that symmetry is paramount, is to keep the same local symmetries plus four-dimensional Poincar´e invariance, but to allow a more general ﬁeld content — that is, two-dimensional ﬁelds in diﬀerent representations of the world-sheet and spacetime symmetry groups. These ideas are both important and we will return to them at various times. A further idea is to enlarge the local symmetry of the theory; we will pursue this idea in volume two. 1.3

The open string spectrum

We discuss the spectrum of the open string in this section and that of the closed in the next, using light-cone gauge to eliminate the diﬀ×Weyl redundancy. This gauge hides the covariance of the theory, but it is the quickest route to the spectrum and reveals important features like the critical dimension and the existence of massless gauge particles. The emphasis here is on the results, not the method, so the reader should not dwell on ﬁne points. A systematic treatment using the covariant conformal gauge will begin in the next chapter. Deﬁne light-cone coordinates in spacetime: x± = 2−1/2 (x0 ± x1 ) ,

xi , i = 2, . . . , D − 1 .

(1.3.1)

We are using lower case for the spacetime coordinates xµ and upper case for the associated world-sheet ﬁelds X µ (σ, τ). In these coordinates the metric is aµ bµ = −a+ b− − a− b+ + ai bi , a− = −a , +

−

a+ = −a ,

(1.3.2a) i

ai = a .

(1.3.2b)

We will set the world-sheet parameter τ at each point of the world-sheet to be equal to the spacetime coordinate x+ . One might have thought to try τ = x0 but this does not lead to the same simpliﬁcations. So x+ will play the role of time and p− that of energy. The longitudinal variables x− and

1.3 The open string spectrum

17

p+ are then like spatial coordinates and momenta, as are the transverse xi and pi . We start by illustrating the procedure in the case of the point particle, . Fix the parameterization of the world-line by using the action Spp X + (τ) = τ .

(1.3.3)

The action then becomes 1 ˙ − + η −1 X ˙ iX ˙ i − ηm2 . Spp = dτ −2η −1 X 2 ˙ µ are The canonical momenta pµ = ∂L/∂X

(1.3.4)

p− = −η −1 ,

(1.3.5)

˙i . pi = η −1 X

Also recall the metric pi = pi and p+ = −p− . The Hamiltonian is ˙ − + pi X ˙i − L H = p− X pi pi + m2 . (1.3.6) 2p+ ˙ + because X + is not a dynamical variable Note that there is no term p+ X in the gauge-ﬁxed theory. Also, η˙ does not appear in the action and so the momentum pη vanishes. Since η = −1/p− it is clear that we should not treat η as an independent canonical coordinate; in a more complete treatment we would justify this by appealing to the theory of systems with constraints. To quantize, impose canonical commutators on the dynamical ﬁelds, =

[pi , X j ] = −iδi j ,

[p− , X − ] = −i .

(1.3.7)

The momentum eigenstates |k− form a complete set. The remaining momentum component p+ is determined in terms of the others as follows. The gauge choice relates τ and X + translations, so H = −p+ = p− . The relative sign between H and p+ arises because the former is active and the latter passive. Then eq. (1.3.6) becomes the relativistic mass-shell condition, and we have obtained the spectrum of a relativistic scalar. Now turn to the open string, taking again the coordinate region −∞ < τ < ∞ and 0 ≤ σ ≤ . Again we must make a gauge choice to ﬁx the redundancies in the Polyakov action, and by a good choice we can also make the equations of motion simple. Set , ki

X+ = τ , ∂σ γσσ = 0 , det γab = −1 .

(1.3.8a) (1.3.8b) (1.3.8c)

That is, we choose light-cone gauge for the world-sheet time coordinate and impose two conditions on the metric. This is three conditions for

18

1 A ﬁrst look at strings

three local symmetries (the choice of two world-sheet coordinates plus the Weyl scaling). Let us see how to reach the gauge (1.3.8). First choose τ according to (1.3.8a). Now notice that f = γσσ (− det γab )−1/2 transforms as f dσ = fdσ

(1.3.9)

under reparameterizations of σ where τ is left ﬁxed. Thus we deﬁne an invariant length fdσ = dl. Deﬁne the σ coordinate of a point to be proportional to its invariant distance dl from the σ = 0 endpoint; the constant of proportionality is determined by requiring that the coordinate of the right-hand endpoint remain at σ = . In this coordinate system f = dl/dσ is independent of σ. It still depends on τ; we would like the parameter region 0 ≤ σ ≤ to remain unchanged, so there is not enough freedom to remove the τ-dependence. Finally, make a Weyl transformation to satisfy condition (1.3.8c). Since f is Weyl-invariant, ∂σ f still vanishes. With condition (1.3.8c) this implies that ∂σ γσσ = 0, and so condition (1.3.8b) is satisﬁed as well.4 We can solve the gauge condition (1.3.8c) for γττ (τ, σ), and γσσ is independent of σ, so the independent degrees of freedom in the metric are now γσσ (τ) and γστ (τ, σ). In terms of these, the inverse metric is

γ ττ γ τσ γ στ γ σσ

=

−γσσ (τ) γτσ (τ, σ) −1 (τ)(1 − γ 2 (τ, σ)) γτσ (τ, σ) γσσ τσ

.

(1.3.10)

The Polyakov Lagrangian then becomes 1 L=− 4πα

0

dσ γσσ (2∂τ x− − ∂τ X i ∂τ X i )

−1 2 −2γστ (∂σ Y − − ∂τ X i ∂σ X i ) + γσσ (1 − γτσ )∂σ X i ∂σ X i .

(1.3.11) Here we have separated X − (τ, σ) into two pieces x− (τ) and Y − (τ, σ), the ﬁrst being the mean value of X − at given τ and the second having a mean value of zero. That is,

1 dσX − (τ, σ) , x (τ) = 0 Y − (τ, σ) = X − (τ, σ) − x− (τ) . −

4

(1.3.12a) (1.3.12b)

We have shown how to assign a deﬁnite (τ, σ) value to each point on the world-sheet, but we have not shown that this is a good gauge — that diﬀerent points have diﬀerent coordinates. Already for the point particle this is an issue: τ is a good coordinate only if the world-line does not double back through an x+ -hyperplane. For the string we need this and also that f is positive everywhere. It is not hard to show that for allowed classical motions, in which all points travel at less than the speed of light, these conditions are met. In the quantum theory, one can show for the point particle that the doubled-back world-lines give no net contribution. Presumably this can be generalized to the string, but this is one of those ﬁne points not to dwell on.

19

1.3 The open string spectrum

The ﬁeld Y − does not appear in any terms with time derivatives and so is nondynamical. It acts as a Lagrange multiplier, constraining ∂σ2 γτσ to vanish (extra ∂σ because Y − has zero mean). Note further that in the gauge (1.3.8) the open string boundary condition (1.2.28) becomes γτσ ∂τ X µ − γττ ∂σ X µ = 0

at σ = 0, .

(1.3.13)

For µ = + this gives γτσ = 0

at σ = 0,

(1.3.14)

and since ∂σ2 γτσ = 0, then γτσ vanishes everywhere. For µ = i, the boundary condition is ∂σ X i = 0

at σ = 0, .

(1.3.15)

After imposing the gauge conditions and taking into account the Lagrange multiplier Y − , we can proceed by ordinary canonical methods. The system is reduced to the variables x− (τ), γσσ (τ) and the ﬁelds X i (τ, σ). The Lagrangian is

1 − i i −1 i i γ ∂ x + dσ γ ∂ X ∂ X − γ ∂ X ∂ X . (1.3.16) σσ τ σσ τ τ σ σσ σ 2πα 4πα 0 The momentum conjugate to x− is

L=−

p− = −p+ =

∂L γσσ . =− − ∂(∂τ x ) 2πα

(1.3.17)

As with η in the particle example, γσσ is a momentum and not a coordinate. The momentum density conjugate to X i (τ, σ) is Πi =

δL p+ 1 i γ ∂ X = = ∂τ X i . σσ τ δ(∂τ X i ) 2πα

The Hamiltonian is then −

H = p− ∂τ x − L + = 4πα p+

0

0

(1.3.18)

dσ Πi ∂τ X i

1 dσ 2πα Π Π + ∂σ X i ∂σ X i 2πα

i

i

.

(1.3.19)

This is just the Hamiltonian for D − 2 free ﬁelds X i , with p+ a conserved quantity. The equations of motion are ∂H H ∂H = + , ∂ τ p+ = − = 0 , ∂p− p ∂x δH δH c 2 i = 2πα cΠi , ∂τ Πi = − i = ∂ X , ∂τ X i = δΠi δX 2πα σ implying the wave equation ∂τ x− =

∂τ2 X i = c2 ∂σ2 X i

(1.3.20a) (1.3.20b)

(1.3.21)

20

1 A ﬁrst look at strings

with velocity c = /2πα p+ . We will not develop the interactions in the light-cone formalism, but for these it is useful to choose the coordinate length for each string to be proportional to p+ , so that c = 1. Since p+ is positive and conserved, the total string length is then also conserved. The transverse coordinates satisfy a free wave equation so it is useful to expand in normal modes. In fact X ± also satisfy the free wave equation. For X + this is trivial, while for X − it requires a short calculation. The general solution to the wave equation with boundary condition (1.3.15) is

∞ pi πincτ πnσ 1 i αn exp − . (1.3.22) cos X (τ, σ) = x + + τ + i(2α )1/2 p n=−∞ n i

i

n=0

Reality of X i requires αi−n = (αin )† . We have deﬁned the center-of-mass variables 1 i x (τ) = dσ X i (τ, σ) , (1.3.23a) 0 p+ dσ Πi (τ, σ) = dσ ∂τ X i (τ, σ) , (1.3.23b) pi (τ) = 0 0 which are the average position and the total momentum. These are the Heisenberg operators; the Schr¨ odinger operators xi ≡ xi (0) and pi ≡ pi (0) appear in the expansion (1.3.22). To quantize, impose the equal time canonical commutation relations [x− , p+ ] = iη −+ = −i , [X i (σ), Πj (σ )] = iδ ij δ(σ − σ ) ,

(1.3.24a) (1.3.24b)

with all other commutators between the independent variables vanishing. In terms of the Fourier components, [xi , pj ] = iδ ij , [αim , αjn ] = mδ ij δm,−n .

(1.3.25a) (1.3.25b)

For each m and i the modes satisfy a harmonic oscillator algebra with nonstandard normalization αim ∼ m1/2 a ,

αi−m ∼ m1/2 a† ,

m>0,

(1.3.26)

where [a, a† ] = 1. The oscillators are labeled by the direction of oscillation i and the harmonic m. The state |0; k , where k = (k + , k i ), is deﬁned to be annihilated by the lowering operators and to be an eigenstate of the center-of-mass momenta, p+ |0; k = k + |0; k , pi |0; k = k i |0; k , αim |0; k = 0 , m > 0 .

(1.3.27a) (1.3.27b)

21

1.3 The open string spectrum

A general state can be built by acting on |0; k with the raising operators, |N; k =

D−1 ∞ i=2

(αi−n )Nin |0; k . (nNin Nin !)1/2 n=1

(1.3.28)

That is, the independent states can be labeled by the center-of-mass momenta k + and k i , and by the occupation numbers Nin for each mode (i, n), where i = 2, . . . , D − 1 and n = 1, . . . , ∞. The center-of-mass momenta are just the degrees of freedom of a point particle, while the oscillators represent an inﬁnite number of internal degrees of freedom. Every choice of these occupation numbers corresponds, from the spacetime point of view, to a diﬀerent particle or spin state. Note that the states (1.3.28) form the Hilbert space H1 of a single open string, and in particular the state |0; 0 is the ground state of a single string with zero momentum, not the zero-string vacuum state. We will call the latter |vacuum to make this clear. The various operators appearing above do not create or destroy strings but act within the space of states of a single string. The n-string Hilbert space Hn would be formed as the product of n copies of the space (1.3.28); the wavefunction must be symmetrized because, as we will see, all these states have integer spin. The full Hilbert space of the string theory, at least in the free limit that we are considering here, is then the sum H = |vacuum ⊕ H1 ⊕ H2 ⊕ . . . .

(1.3.29)

Inserting the mode expansion (1.3.22) into the Hamiltonian (1.3.19) gives pi p i 1 H= ++ + 2p 2p α

∞

αi−n αin

+A

.

(1.3.30)

n=1

In the Hamiltonian H, the order of operators is ambiguous. We have put the lowering operators on the right and the raising operators on the left, and included an unknown constant A from the commutators. In a careful treatment of light-cone quantization, this constant is determined as follows. The choice of light-cone gauge has obscured the Lorentz invariance of the theory. It is necessary to check this by ﬁnding the operators M µν that generate Lorentz transformations, and verifying that they have the correct algebra with pµ and with each other. One ﬁnds that this is the case only for the speciﬁc value A = −1, and only if the spacetime dimensionality is precisely D = 26. We do not wish to spend this much time on light-cone quantization, so we will try to show that the result is plausible without a systematic treatment. We assure the reader that we will obtain the values of A and D honestly in the conformal gauge approach, which will also give us insight

22

1 A ﬁrst look at strings

into why Lorentz invariance would be lost in the light-cone approach for the wrong A or D. First, we assert that the operator ordering constant in the Hamiltonian for a free ﬁeld comes from summing the zero-point energies of each oscillator mode, 12 ω for a bosonic ﬁeld like X µ . Equivalently, it always works out that the natural operator order is averaged, 12 ω(aa† + a† a), which is the same as ω(a† a + 12 ). In H this would give A=

∞ D−2 n, 2 n=1

(1.3.31)

the factor of D − 2 coming from the sum over transverse directions. The zero-point sum diverges. It can be evaluated by regulating the theory and then being careful to preserve Lorentz invariance in the renormalization. This leads to the odd result ∞

n→−

n=1

1 . 12

(1.3.32)

To motivate this, insert a smooth cutoﬀ factor −1/2 exp(−γσσ |kσ |)

(1.3.33) −1/2

into the sum, where kσ = nπ/ and the factor of γσσ is included to make this invariant under σ reparameterizations. The zero-point constant is then A→

∞

D−2 n exp −n(π/2p+ α )1/2 2 n=1

D − 2 2p+ α 1 = − + O() 2 2 π 12

(1.3.34)

.

The cutoﬀ-dependent ﬁrst term is proportional to the length of the and can be canceled by a counterterm in the action proportional to string 2 d σ(−γ)1/2 . In fact, Weyl invariance requires that it be canceled, leaving only the cutoﬀ-independent second term, A=

2−D . 24

(1.3.35)

This ﬁnite remainder is an example of a Casimir energy, coming from the fact that the string has a ﬁnite length. As for the point particle, p− = H, so

1 2−D m = 2p H − p p = N + α 24 2

+

i i

,

(1.3.36)

1.3 The open string spectrum

23

where N is the level N=

D−1 ∞

nNin .

(1.3.37)

i=2 n=1

The mass of each state is thus determined in terms of the level of excitation. Now let us look at some of the light string states. The lightest is 2−D . (1.3.38) 24α The mass-squared is negative if D > 2: the state is a tachyon. In ﬁeld theory the potential energy for a scalar ﬁeld is 12 m2 φ2 , so the negative mass-squared means that the no-string ‘vacuum’ is actually unstable, like the symmetric state in a spontaneously broken theory. It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known. Starting with our study of the superstring in volume two, we will see that there are tachyon-free string theories. For now, the simplest way to proceed is to ignore this instability and use the bosonic string theory as our model for developing string technology. Occasionally we will encounter divergences due to the tachyon, but these will not interfere with our purpose. The lowest excited states of the string are obtained by exciting one of the n = 1 modes once: 26 − D . (1.3.39) αi−1 |0; k , m2 = 24α Lorentz invariance now requires a speciﬁc value of D as follows. The analysis of spin is diﬀerent for massive and massless particles. For a massive particle, one goes to the rest frame pµ = (m, 0, . . . , 0). The internal states then form a representation of the spatial rotation group S O(D − 1). For a massless particle there is no rest frame; choose the frame pµ = (E, E, 0, . . . , 0). The S O(D − 2) acting on the transverse directions leaves pµ invariant, and the internal states form a representation of this smaller group. This is familiar from D = 4: massive particles are labeled by spin j, the S O(3) representation, and so have 2j + 1 states. Massless particles are labeled by their helicity λ, which is their eigenvalue under the single generator of S O(2). Lorentz invariance alone thus requires only one state, though CP T symmetry takes λ to −λ and so requires two states for λ = 0. In D dimensions, a massive vector particle thus has D − 1 spin states while a massless vector need have only D − 2 states. At the ﬁrst level we found only the D − 2 states αi−1 |0; k , so these must be massless and |0; k ,

m2 =

A = −1,

D = 26 .

(1.3.40)

This is a striking and important result: the spectrum is Lorentz-invariant only if the number of spacetime dimensions is D = 26. The classical theory

24

1 A ﬁrst look at strings

is Lorentz-invariant for any D, but there is an anomaly — the symmetry is not preserved by the quantization except when D = 26. In later chapters we will obtain a deeper understanding of how this comes about and what it implies. We conclude this section with a few more comments about spin. The light-cone quantization singles out two directions and leaves manifest only the S O(D − 2) that acts in the transverse directions. The spin generators for the transverse directions are S = −i ij

∞ 1

n n=1

(αi−n αjn − αj−n αin ) ;

(1.3.41)

antisymmetry on ij together with Lorentz invariance allows no zero-point constant. For a massless particle, the full S O(D −2) spin symmetry is made manifest by choosing the momentum to lie along the 1-direction singled out in the quantization. For a massive particle, only an S O(D − 2) subgroup of the S O(D − 1) spin symmetry will be manifest in light-cone quantization, but this is still quite useful. For example, the (D−1)-dimensional vector representation of S O(D − 1) breaks up into an invariant and a (D−2)-vector under the S O(D − 2) acting on the transverse directions, v = (v 1 , 0, . . . , 0) + (0, v 2 , . . . , v D−1 ) .

(1.3.42)

Thus, if a massive particle is in the vector representation of S O(D − 1), we will see a scalar and a vector when we look at the transformation properties under S O(D − 2). This idea extends to any representation: one can always reconstruct the full S O(D − 1) spin representation from the behavior under S O(D − 2). The higher excited states of the string, which are massive, do form full representations of S O(D − 1). One can check this by hand for a few levels, and in a more complete analysis it follows from the existence of the full set of Lorentz generators. We will show this indirectly in the coming chapters, by ﬁnding a consistent covariant quantization. At level N, the maximum eigenvalue of a given spin component, say S 23 , is N, obtained by acting N times with α2−1 + iα3−1 . Thus, S 23 ≤ 1 + α m2 .

(1.3.43)

The slope in this inequality is known as the Regge slope. Meson resonances obey a linear relation of this form, with α ∼ (1 GeV)−2 . For this and other reasons, string theory was originally proposed in 1970 as a theory of the strong interaction. Within a few years, however, the S U(3) gauge theory QCD was discovered to be the correct theory, with string-like behavior only in certain limits.

1.4 Closed and unoriented strings

25

Now that we are considering strings as the uniﬁed theory of particle physics, gravity, and quantum mechanics, α will be of the order of the natural scale determined by the fundamental constants of gravity and quantum mechanics, MP−2 . In particular, masses that are not zero in string theory are of order MP . This is so large compared to experimentally accessible energy scales that these particles appear only in virtual states. Thus, we will be especially concerned with the massless string spectrum, since this must include all the particles of the Standard Model. Of course, most known particles are massive, but these masses are so small compared to MP that they are zero to ﬁrst approximation and become non-zero due to small symmetry-breaking eﬀects. Some of the interest in string theory has been driven by the possibility that the strong interaction theory QCD is equivalent to a string theory, at least in some approximation. There has been cross-fertilization between this idea and the idea of strings as a fundamental theory. The connection between QCD and string theory is outside the focus of this book, but the recent developments described in chapter 14 have also given new insight in this direction. 1.4

Closed and unoriented strings

The light-cone quantization of the closed string is quite similar to that of the open string. Again impose the gauge conditions (1.3.8). In the open string, these determined the gauge completely. In the closed string there is still some extra coordinate freedom, σ = σ + s(τ) mod ,

(1.4.1)

because the point σ = 0 can be chosen anywhere along the string. Most of this remaining freedom can be ﬁxed by the additional gauge condition γτσ (τ, 0) = 0 .

(1.4.2)

That is, the line σ = 0 is orthogonal to the lines of constant τ. This determines the line σ = 0 except for an overall σ-translation. Conditions (1.3.8) and (1.4.2) thus ﬁx all of the gauge freedom except for τ-independent translations of σ, σ = σ + s mod .

(1.4.3)

We will deal with this extra gauge freedom later. The analysis is now parallel to that for the open string. The Lagrangian, canonical momenta, Hamiltonian, and equation of motion are as in the open string, eqs. (1.3.16)–(1.3.21). The general periodic solution to the

26

1 A ﬁrst look at strings

equation of motion is

α 1/2 pi τ + i p+ 2 ∞ i α 2πin(σ − cτ) 2πin(σ + cτ) α˜in n × + . exp − exp n n n=−∞

X i (τ, σ) = xi +

n=0

(1.4.4) In the closed string there are two independent sets of oscillators, αin and ˜αin , corresponding to left-moving and right-moving waves along the string. In the open string the boundary condition at the endpoints tied these together. The independent degrees of freedom are again the transverse oscillators and the transverse and longitudinal center-of-mass variables, αin , α˜in , xi , pi , x− , p+ ,

(1.4.5)

with canonical commutators

kµ

[x− , p+ ] = −i , [xi , pj ] = iδ ij , [αim , αjn ] = mδ ij δm,−n ,

(1.4.6a) (1.4.6b) (1.4.6c)

[˜αim , ˜αjn ] = mδ ij δm,−n .

(1.4.6d)

Starting from the state |0, 0; k , which has center-of-mass momentum and is annihilated by αim s and ˜αim s for m > 0, the general state is ˜ k = |N, N;

D−1 ∞

i=2 n=1

˜

(αi−n )Nin (˜αi−n )Nin |0, 0; k . ˜ in !)1/2 (nNin Nin !nN˜ in N

(1.4.7)

The mass formula is m2 = 2p+ H − pi pi

∞ 2 i i i i ˜ = (α α + ˜α−n ˜αn ) + A + A α n=1 −n n 2 ˜ + A + A) ˜ . (N + N (1.4.8) α We have broken up the level and zero-point constant into the part from the right-moving modes and the part from the left. Summing zero-point energies again gives =

2−D . (1.4.9) 24 There is one further restriction on the state due to the remaining gauge freedom, the σ-translations (1.4.3). The physical spectrum is obtained ˜= A=A

27

1.4 Closed and unoriented strings

by restricting to gauge-invariant states. The operator that generates the σ-translations is P =−

0

dσ Πi ∂σ X i

∞ 2π ˜ =− (αi αi − ˜αi−n ˜αin ) + A − A n=1 −n n

2π ˜ . (N − N) States must therefore satisfy =−

(1.4.10)

˜ . N=N

(1.4.11)

The lightest closed string state is 2−D , 6α which is again a tachyon. The ﬁrst excited states are |0, 0; k ,

m2 =

(1.4.12)

26 − D . (1.4.13) 6α As with the open string, these states do not add up to complete representations of S O(D − 1) and so this level must be massless. Thus, ˜ = −1 , D = 26 . A=A (1.4.14) αi−1 ˜αj−1 |0, 0; k ,

m2 =

The states (1.4.13) transform as a 2-tensor under S O(D − 2). This is a reducible representation: it decomposes into a symmetric traceless tensor, an antisymmetric tensor, and a scalar. That is, any tensor eij can be decomposed

1 ij 2 1 ij 1 δ ij ekk + δ ij ekk , (1.4.15) e + eji − e − eji + e = 2 D−2 2 D−2 and the three separate terms do not mix under rotations. ˜ in are independent except for the N = N ˜ At any mass level, Nin and N 2 constraint. Thus, the closed string spectrum at m = 4(N − 1)/α is the product of two copies of the m2 = (N − 1)/α level of the open string. Let us now discuss some general issues. The two-dimensional diﬀ invariance removed two families of normal modes from the spectrum. If we tried to make a covariant theory without this invariance, we would have to generalize the transverse commutator (1.3.25) to ij

[αµm , ανn ] = mη µν δm,−n .

(1.4.16)

Lorentz invariance forces the timelike oscillators to have the wrong-sign commutator. A state with an odd number of timelike excitations will then have a negative norm. This is inconsistent with quantum mechanics,

28

1 A ﬁrst look at strings

since the norm is a probability. In fact a theory based on (1.4.16) came before the string theory we have described, and the latter was discovered by requiring that the negative norm states never be produced in physical processes. The commutator (1.4.16) arises in a covariant quantization of string theory (which we shall use beginning in the next chapter), with the coordinate invariance appearing through constraints on the spectrum to eliminate the unphysical states. The existence of a massless vector in the open string theory and of the massless symmetric and antisymmetric tensors in the closed string theory is striking. General principles require that a massless vector couple to a conserved current and therefore that the theory have a gauge invariance. This is our ﬁrst example of the massless gauge particles that are present in all fundamental string theories. Actually, this particular gauge boson, call it the photon, is not so interesting, because the gauge group is only U(1) (there is only one photon) and because it turns out that all particles in the theory are neutral. In chapter 6 we will discuss a simple generalization of open string theory, adding Chan–Paton degrees of freedom at the string endpoints, which leads to U(n), S O(n), and S p(n) gauge groups. Similarly, a massless symmetric tensor particle must couple to a conserved symmetric tensor source. The only such source is the energy-momentum tensor; some additional possibilities arise in special cases, but these are not relevant here. Coupling to this in a consistent way requires the theory to have spacetime coordinate invariance. Thus the massless symmetric tensor is the graviton, and general relativity is contained as one small piece of the closed string theory. The massless antisymmetric tensor is known as a 2-form gauge boson. As we will see in section 3.7, there is also a local spacetime symmetry associated with it. These spacetime gauge and coordinate invariances were not evident in our starting point, the Nambu–Goto or Polyakov theory. We have certainly discovered them in a backwards way: adding up zero-point energies, requiring a Lorentz-invariant spectrum, and citing general results about interactions of massless particles. In gauge theory and in general relativity, one starts with a spacetime symmetry principle. In string theory, too, it would seem that we should ﬁrst ﬁgure out what the full spacetime symmetry is, and use this to deﬁne the theory. There are various attempts in this direction, but a complete picture has not yet emerged. For now, let us note one thing: we have observed that the local world-sheet symmetry diﬀ removes the unphysical timelike and longitudinal normal modes which would lead to negative norm string states. For example, the open string photon would otherwise have D states rather than D−2. From the spacetime point of view, it is the local spacetime invariances that remove these same unphysical states. So there is at least some connection between the local symmetries of the world-sheet and those of spacetime.

1.4 Closed and unoriented strings

29

Anticipating our interest in four dimensions, let us look at the massless string states obtained by exciting only oscillators in the 2- and 3-directions, and classifying them by the helicity λ = S 23 . One ﬁnds that the photon has λ = ±1 and the graviton λ = ±2, justifying the names. The closed string scalar and antisymmetric tensor each give rise to a λ = 0 state. These are called the dilaton and the axion, respectively, and we will discuss their physics at various points. We conclude this section with a discussion of one other generalization, the unoriented string theory. The theories we have been discussing are oriented string theories. We have not considered the coordinate transformation σ = − σ ,

τ = τ ,

(1.4.17)

which changes the orientation (handedness) of the world-sheet. This symmetry is generated by the world-sheet parity operator Ω. Carrying out (1.4.17) twice gives the identity, so Ω2 = 1 and the eigenvalues of Ω are ±1. From the mode expansions (1.3.22) and (1.4.4) we see that Ωαin Ω−1 = (−1)n αin

(1.4.18)

Ωαin Ω−1 = α˜in , Ω˜αin Ω−1 = αin

(1.4.19a) (1.4.19b)

in the open string and

in the closed string. We deﬁne the phase of Ω by ﬁxing Ω = +1 for the ground states |0; k and |0, 0; k ; later we will see that this choice is required in order for Ω to be conserved by interactions. Then Ω|N; k = (−1)N |N; k , ˜ k = |N, ˜ N; k . Ω|N, N;

(1.4.20a) (1.4.20b)

There are consistent interacting string theories, the unoriented theories, in which only the Ω = +1 states are kept. Focusing on the massless states, the open string photon has Ω = −1 and is absent in the unoriented theory. Acting on the massless tensor states in the closed string, the parity operator Ω takes eij to eji , so the graviton and dilaton are present in the unoriented theory while the antisymmetric tensor is not. Notice that both the open and closed string tachyons survive in the unoriented theory; we will have to work harder to remove these. Let us mention two other constraints that will emerge from the study of the interactions. First, it is possible to have a consistent theory with only closed strings, or with closed and open strings, but not with open strings alone: closed strings can always be produced in the scattering of open strings. Second, oriented or unoriented open strings can only couple to

30

1 A ﬁrst look at strings

closed strings of the same type. We list the possible combinations together with their massless spectra, with Gµν representing the graviton, Bµν the antisymmetric tensor, Φ the dilaton, and Aµ the photon: 1. Closed oriented bosonic string: Gµν , Bµν , Φ . 2. Closed unoriented bosonic string: Gµν , Φ . 3. Closed plus open oriented bosonic string: Gµν , Bµν , Φ, Aµ . 4. Closed plus open unoriented bosonic string: Gµν , Φ . All of these have the graviton and dilaton, as well as the tachyon. In chapter 6 we will discuss more general open string theories, with Chan– Paton degrees of freedom at the endpoints. The massless oriented open strings will then be U(n) gauge bosons, and the massless unoriented open strings will be S O(n) or S p(n) gauge bosons. Exercises 1.1 (a) Show that in the nonrelativistic limit the action Spp has the usual nonrelativistic form, kinetic energy minus potential energy, with the potential energy being the rest mass. (b) Show that for a string moving nonrelativistically, the Nambu–Goto action reduces to a kinetic term minus a potential term proportional to the length of the string. Show that the kinetic energy comes only from the transverse velocity of the string. Calculate the mass per unit length, as determined from the potential term and also from the kinetic term. 1.2 Show that classical equations of motion imply that the ends of the open string move at the speed of light. 1.3 For world-sheets with boundary, show that

1 1 dτ dσ (−γ)1/2 R + ds k 4π M 2π ∂M is Weyl-invariant. Here ds is the proper time along the boundary in the metric γab , and k is the geodesic curvature of the boundary, χ=

k = ±ta nb ∇a tb , where ta is a unit vector tangent to the boundary and na is an outward pointing unit vector orthogonal to ta . The upper sign is for a Lorentzian world-sheet and the lower sign for a Euclidean world-sheet. 1.4 Show that the open string states at levels m2 = 1/α and 2/α form complete representations of S O(D − 1). You will need to work out the S O(D − 2) content of various symmetric and antisymmetric tensor repre-

Exercises

31

sentations of S O(D − 1). You can carry this to higher levels if you like, but you will need tensors of mixed symmetry and Young tableaux. 1.5 Extend the sum (1.3.32) to the ‘twisted’ case ∞

(n − θ)

n=1

with θ a constant. That is, kσ = (n − θ)π/. The answer is given in eq. (2.9.19). You should ﬁnd that the cutoﬀ-dependent term is independent of θ. In the following exercises one has the usual open or closed string boundary conditions (Neumann or periodic) on X µ for µ = 0, . . . , 24 but a diﬀerent boundary condition on X 25 . Each of these has an important physical interpretation, and will be developed in detail in chapter 8. Find the mode expansion, the mass spectrum, and (for the closed string) the constraint from σ-translation invariance in terms of the occupation numbers. In some cases you need the result of exercise 1.5. 1.6 Open strings with X 25 (τ, 0) = 0 ,

X 25 (τ, ) = y

with y a constant. This is an open string with both ends on D-branes. 1.7 Open strings with X 25 (τ, 0) = 0 ,

∂σ X 25 (τ, ) = 0 .

This is an open string with one end on a D-brane and one end free. 1.8 Closed strings with X 25 (τ, σ + ) = X 25 (τ, σ) + 2πR with R a constant. This is a winding string in toroidal (periodic) compactiﬁcation. In this case p25 must be a multiple of 1/R. 1.9 Closed strings with X 25 (τ, σ + ) = −X 25 (τ, σ) . This is a twisted string in orbifold compactiﬁcation.

2 Conformal ﬁeld theory

In this chapter we develop a number of necessary ideas and techniques from the world-sheet quantum ﬁeld theory, including the operator product expansion, conformal invariance, the Virasoro algebra, and vertex operators. The focus is on conformally invariant ﬁeld theory in two ﬂat dimensions; as we will see in the next chapter, this is what we are left with after ﬁxing the local symmetries of the string world-sheet. 2.1

Massless scalars in two dimensions

We will start with the example of D free scalar ﬁelds in two dimensions, X µ (σ 1 , σ 2 ). We will refer to these two dimensions as the world-sheet, anticipating the application to string theory. The action is 1 S= 4πα

d2 σ ∂1 X µ ∂1 X µ + ∂2 X µ ∂2 X µ .

(2.1.1)

This is the Polyakov action (1.2.13), except that the world-sheet metric γab has been replaced with a ﬂat Euclidean metric δab , signature (+, +). The overall sign change of the action is a result of the Euclidean convention (A.1.31). As we will see, most string calculations are carried out on a Euclidean world-sheet. At least for ﬂat metrics, the relation between Euclidean and Minkowski amplitudes is given by a standard analytic continuation, explained in the appendix. In fact, the results of the present chapter apply equally to a Minkowski world-sheet, and in the ﬁrst seven sections all equations make sense if σ 2 is replaced with iσ 0 . For the index µ we still take the ﬂat Minkowski metric. It is straightforward to quantize the action (2.1.1) canonically, ﬁnding the spectrum, vacuum expectation values, and so on. We have done essentially this in chapter 1, after having gone to light-cone gauge. Here we will take a somewhat diﬀerent route, developing ﬁrst various local properties such 32

2.1 Massless scalars in two dimensions

33

as equations of motion, operator products, Ward identities, and conformal invariance, before working our way around to the spectrum. It will be eﬃcient for us to use the path integral formalism. This is reviewed in the appendix. We will be using the path integral representation primarily to derive operator equations (to be deﬁned below); these can also be derived in a Hilbert space formalism. It is very useful to adopt complex coordinates z = σ 1 + iσ 2 ,

¯z = σ 1 − iσ 2 .

(2.1.2)

We will use a bar for the complex conjugates of z and other simple variables, and a star for the complex conjugates of longer expressions. Deﬁne also 1 ∂z = (∂1 − i∂2 ) , 2

1 ∂¯z = (∂1 + i∂2 ) . 2

(2.1.3)

These derivatives have the properties ∂z z = 1 ,

∂z ¯z = 0 ,

∂¯z z = 0 ,

∂¯z ¯z = 1 .

(2.1.4)

It is conventional to abbreviate ∂z to ∂ and ∂¯z to ∂¯ when this is not ambiguous. For a general vector v a , deﬁne in the same way v z = v 1 +iv 2 ,

1 vz = (v 1 −iv 2 ) , 2

v¯z = v 1 −iv 2 ,

1 v¯z = (v 1 +iv 2 ) . (2.1.5) 2

For the indices 1, 2 the metric is the identity and we do not distinguish between upper and lower, while the complex indices are raised and lowered with gz¯z = g¯z z =

1 , 2

gzz = g¯z¯z = 0 ,

g z¯z = g¯z z = 2 ,

g zz = g¯z¯z = 0 . (2.1.6)

Note also that d2 z = 2dσ 1 dσ 2

(2.1.7)

with the factor of 2 from the Jacobian,1 and that d2 z | det g|1/2 = dσ 1 dσ 2 . We deﬁne

so that δ 2 (z, ¯z ) =

1

d2 z δ 2 (z, ¯z ) = 1

1 1 2 2 δ(σ )δ(σ ).

(2.1.8)

Another useful result is the divergence

This diﬀers from much of the literature, where d2 z is deﬁned as dσ 1 dσ 2 . Correlated with this, the 1 2 z ). 2 is omitted from the deﬁnition of δ (z, ¯

34

2 Conformal ﬁeld theory

theorem in complex coordinates,

R

d2 z (∂z v z + ∂¯z v¯z ) = i

∂R

(v z d¯z − v¯z dz) ,

(2.1.9)

where the contour integral circles the region R counterclockwise. In this notation the action is S=

1 2πα

¯ µ d2 z ∂X µ ∂X

(2.1.10)

and the classical equation of motion is ¯ µ (z, ¯z ) = 0 . ∂∂X

(2.1.11)

The notation X µ (z, ¯z ) may seem redundant, since the value of z determines the value of ¯z , but it is useful to reserve the notation f(z) for ﬁelds whose equation of motion makes them analytic (equivalently holomorphic) functions of z. Writing the equation of motion as µ ¯ ¯ µ ) = ∂(∂X )=0, ∂(∂X

(2.1.12)

¯ µ is antiholomorphic it follows that ∂X µ is holomorphic and that ∂X ¯ µ (¯z ). (holomorphic in ¯z ), hence the notations ∂X µ (z) and ∂X 2 0 Under the Minkowski continuation σ = iσ , a holomorphic ﬁeld becomes a function only of σ 0 − σ 1 and an antiholomorphic ﬁeld a function only of σ 0 + σ 1 . We thus use as synonyms holomorphic = left-moving , antiholomorphic = right-moving .

(2.1.13a) (2.1.13b)

This terminology is chosen to have maximal agreement with the literature, though for it to hold literally we would need to draw σ 1 increasing from right to left. We will tend to use the Euclidean terms early on and shift to the Minkowski terms as we discuss more of the spacetime physics. Expectation values are deﬁned by the path integral,

F[X] =

[dX] exp(−S )F[X] ,

(2.1.14)

where F[X] is any functional of X, such as a product of local operators. The path integral over X 0 is a wrong-sign Gaussian, so it should be understood to be deﬁned by the analytic continuation X 0 → −iX D . The reader should not be distracted by this; we will discuss it further in the next chapter. We do not normalize F[X] by dividing by 1 . The path integral of a total derivative is zero. This is true for ordinary bosonic path integrals, which can be regarded as the limit of an inﬁnite number of ordinary integrals, as well as for more formal path integrals as

35

2.1 Massless scalars in two dimensions with Grassmann variables. Then

δ exp(−S ) δXµ (z, ¯z ) δS = − [dX] exp(−S ) δXµ (z, ¯z )

0=

[dX]

=− =

δS δXµ (z, ¯z )

1 ¯ µ ¯ ∂ ∂X (z, z ) . πα

(2.1.15)

The same calculation goes through if we have arbitrary additional insertions ‘. . .’ in the path integral, as long as none of these additional insertions is at z. Thus

¯ µ (z, ¯z ) . . . ∂∂X

=0.

(2.1.16)

We can regard the additional insertions as preparing arbitrary initial and ﬁnal states (or we could do the same thing with boundary conditions). The path integral statement (2.1.16) is thus the same as the statement in the Hilbert space formalism that ˆ µ (z, ¯z ) = 0 ∂∂¯X

(2.1.17)

ˆ µ (z, ¯z ). Thus we refer holds for all matrix elements of the operator X to relations that hold in the sense (2.1.16) as operator equations. The statement (2.1.17) is Ehrenfest’s theorem that the classical equations of motion translate into operator equations. The notation ‘. . .’ in the path integral (2.1.16) implicitly stands for insertions that are located away from z, but it is interesting to consider also the case in which there is an insertion that might be coincident with z:

[dX]

0=

=

δ exp(−S )X ν (z , ¯z ) δXµ (z, ¯z )

[dX] exp(−S ) η µν δ 2 (z − z , ¯z − ¯z ) +

= η µν δ 2 (z − z , ¯z − ¯z ) +

1 µ ν ¯ ¯ ∂ ∂ X (z, z )X (z , z ) ¯ z z πα

1 ∂z ∂¯z X µ (z, ¯z )X ν (z , ¯z ) . πα

(2.1.18)

That is, the equation of motion holds except at coincident points. Again this goes through with arbitrary additional insertions ‘. . .’ in the path integral, as long as none of these additional ﬁelds is at (z, ¯z ) or (z , ¯z ): µ 1 ν µν 2 ¯ ¯ ¯ ¯ ∂ ∂ X (z, z )X (z , z ) . . . = −η δ (z − z , z − z ) . . . . (2.1.19) z ¯z πα

36

2 Conformal ﬁeld theory

Thus, 1 ∂z ∂¯z X µ (z, ¯z )X ν (z , ¯z ) = −η µν δ 2 (z − z , ¯z − ¯z ) (2.1.20) πα holds as an operator equation. In the Hilbert space formalism, the product in the path integral becomes a time-ordered product, and the delta function comes from the derivatives acting on the time-ordering. This connection is developed further in the appendix. In free ﬁeld theory, it is useful to introduce the operation of normal ordering. Normal ordered operators, denoted : A :, are deﬁned as follows, : X µ (z, ¯z ) : = X µ (z, ¯z ) , : X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) : = X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) +

(2.1.21a) α 2

η µν ln |z12 |2 , (2.1.21b)

where zij = zi − zj .

(2.1.22)

The reader may be familiar with normal ordering deﬁned in terms of raising and lowering operators; these two deﬁnitions will be related later. The point of this deﬁnition is the property ∂1 ∂¯1 : X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) : = 0 .

(2.1.23)

This follows from the operator equation (2.1.20) and the diﬀerential equation ∂∂¯ ln |z|2 = 2πδ 2 (z, ¯z ) .

(2.1.24)

Eq. (2.1.24) is obvious for z = 0 because ln |z|2 = ln z + ln ¯z , and the normalization of the delta function is easily checked by integrating both sides using eq. (2.1.9). 2.2

The operator product expansion

The basic object of interest in string perturbation theory will be the path integral expectation value of a product of local operators,

Ai1 (z1 , ¯z1 ) Ai2 (z2 , ¯z2 ) . . . Ain (zn , ¯zn ) ,

(2.2.1)

where Ai is some basis for the set of local operators. It is particularly important to understand the behavior of this expectation value in the limit that two of the operators are taken to approach one another. The tool that gives a systematic description of this limit is the operator product expansion (OPE), illustrated in ﬁgure 2.1. This states that a product of two

37

2.2 The operator product expansion

z3 z1 z2 z4

Fig. 2.1. Expectation value of a product of four local operators. The OPE gives the asymptotics as z1 → z2 as a series where the pair of operators at z1 and z2 is replaced by a single operator at z2 . The radius of convergence is the distance to the nearest other operator, indicated by the dashed circle.

local operators close together can be approximated to arbitrary accuracy by a sum of local operators, Ai (σ1 )Aj (σ2 ) =

ckij (σ1 − σ2 )Ak (σ2 ) .

(2.2.2)

k

Again this is an operator statement, meaning that it holds inside a general expectation value

Ai (σ1 )Aj (σ2 ) . . . =

ckij (σ1 − σ2 ) Ak (σ2 ) . . .

(2.2.3)

k

as long as the separation between σ1 and σ2 is small compared to the distance to any other operator. The coeﬃcient functions ckij (σ1 − σ2 ), which govern the dependence on the separation, depend on i, j, and k but not on the other operators in the expectation value; the dependence on the latter comes only through the expectation value on the right-hand side of eq. (2.2.3). The terms are conventionally arranged in order of decreasing size in the limit as σ1 → σ2 . This is analogous to an ordinary Taylor series, except that the coeﬃcient functions need not be simple powers and can in fact be singular as σ1 → σ2 , as we will see even in the simplest example. Just as the Taylor series plays a central role in calculus, the OPE plays a central role in quantum ﬁeld theory. We will give now a derivation of the OPE for the X µ theory, using the special properties of free ﬁeld theory. In section 2.9 we will give a derivation for any conformally invariant ﬁeld theory. We have seen that the normal ordered product satisﬁes the naive equation of motion. Eq. (2.1.23) states that the operator product is a harmonic function of (z1 , ¯z1 ). A simple result from the theory of complex variables is that a harmonic function is locally the sum of a holomorphic and an

38

2 Conformal ﬁeld theory

antiholomorphic function. In particular, this means that it is nonsingular as z1 → z2 and can be freely Taylor expanded in z12 and ¯z12 . Thus, α X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) = − η µν ln |z12 |2 + : X ν X µ (z2 , ¯z2 ) : 2 ∞ 1

+ (z12 )k : X ν ∂k X µ (z2 , ¯z2 ) : + (¯z12 )k : X ν ∂¯k X µ (z2 , ¯z2 ) : . k! k=1 (2.2.4) Terms with mixed ∂∂¯ derivatives vanish by the equation of motion. This equation and many others simplify in units in which α = 2, which is the most common convention in the literature. However, several other conventions are also used, so it is useful to keep α explicit. For example, in open string theory equations simplify when α = 12 . Eq. (2.2.4) has the form of an OPE. Like the equation of motion (2.1.23) from which it was derived, it is an operator statement. For an arbitrary expectation value involving the product X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) times ﬁelds at other points, it gives the behavior for z1 → z2 as an inﬁnite series, each term being a known function of z12 and/or ¯z12 times the expectation value with a local operator replacing the pair. OPEs are usually used as asymptotic expansions, the ﬁrst few terms giving the dominant behavior at small separation. Most of our applications will be of this type, and we will often write OPEs as explicit singular terms plus unspeciﬁed nonsingular remainders. The use of ‘∼’ in place of ‘=’ will mean ‘equal up to nonsingular terms.’ In fact, OPEs are actually convergent in conformally invariant ﬁeld theories. This will be very important to us in certain applications: it makes it possible to reconstruct the entire theory from the coeﬃcient functions. As an example, the freeﬁeld OPE (2.2.4) has a radius of convergence in any given expectation value which is equal to the distance to the nearest other insertion in the path integral. The operator product is harmonic except at the positions of operators, and in particular inside the dashed circle of ﬁgure 2.1, and convergence can then be shown by a standard argument from the theory of complex variables. The various operators on the right-hand side of the OPE (2.2.4) involve products of ﬁelds at the same point. Usually in quantum ﬁeld theory such a product is divergent and must be appropriately cut oﬀ and renormalized, but here the normal ordering renders it well-deﬁned. Normal ordering is a convenient way to deﬁne composite operators in free ﬁeld theory. It is of little use in most interacting ﬁeld theories, because these have additional divergences from interaction vertices approaching the composite operator or one another. However, many of the ﬁeld theories that we will be

39

2.2 The operator product expansion

interested in are free, and many others can be related to free ﬁeld theories, so it will be worthwhile to develop normal ordering somewhat further. The deﬁnition of normal ordering for arbitrary numbers of ﬁelds can be given recursively as : X µ1 (z1 , ¯z1 ) . . . X µn (zn , ¯zn ) : = X µ1 (z1 , ¯z1 ) . . . X µn (zn , ¯zn ) + subtractions ,

(2.2.5)

where the sum runs over all ways of choosing one, two, or more pairs of ﬁelds from the product and replacing each pair with 12 α η µi µj ln |zij |2 . For example, : X µ1 (z1 , ¯z1 )X µ2 (z2 , ¯z2 )X µ3 (z3 , ¯z3 ) : = X µ1 (z1 , ¯z1 )X µ2 (z2 , ¯z2 )X µ3 (z3 , ¯z3 ) α µ1 µ2 2 µ3 ln |z12 | X (z3 , ¯z3 ) + 2 permutations . (2.2.6) + η 2 We leave it to the reader to show that the deﬁnition (2.2.5) retains the desired property that the normal ordered product satisﬁes the naive equation of motion. The deﬁnition can be compactly summarized as α

: F := exp

4

d2 z1 d2 z2 ln |z12 |2

δ δ F, µ δX (z1 , ¯z1 ) δXµ (z2 , ¯z2 )

(2.2.7)

where F is any functional of X. This is equivalent to eq. (2.2.5): the double derivative in the exponent contracts each pair of ﬁelds, and the exponential sums over any number of pairs with the factorial canceling the number of ways the derivatives can act. As an example of the use of this formal expression, act on both sides with the inverse exponential to obtain δ α δ d2 z1 d2 z2 ln |z12 |2 :F: F = exp − 4 δX µ (z1 , ¯z1 ) δXµ (z2 , ¯z2 ) = :F: +

contractions ,

(2.2.8)

where a contraction is the opposite of a subtraction: sum over all ways of choosing one, two, or more pairs of ﬁelds from : F : and replacing each pair with − 12 α η µi µj ln |zij |2 . The OPE for any pair of operators can be generated from : F : : G : = : FG : +

cross-contractions

(2.2.9)

for arbitrary functionals F and G of X. The sum now runs over all ways of contracting pairs with one ﬁeld in F and one in G. This can also be written δ δ α 2 2 2 d z1 d z2 ln |z12 | : FG : , : F : : G : = exp − 2 δXFµ (z1 , ¯z1 ) δXGµ (z2 , ¯z2 ) (2.2.10)

40

2 Conformal ﬁeld theory

where the functional derivatives act only on the ﬁelds in F or G respectively. This follows readily from eq. (2.2.7). As an example, : ∂X µ (z)∂Xµ (z) : : ∂ X ν (z )∂ Xν (z ) : = : ∂X µ (z)∂Xµ (z)∂ X ν (z )∂ Xν (z ) : α − 4 · (∂∂ ln |z − z |2 ) : ∂X µ (z)∂ Xµ (z ) : 2 2 α + 2 · η µµ − ∂∂ ln |z − z |2 2 2 Dα 2α ∼ − : ∂ X µ (z )∂ Xµ (z ) : 4 2(z − z ) (z − z )2 2α − : ∂2 X µ (z )∂ Xµ (z ) : . (2.2.11) z − z The second term in the equality comes from the four ways of forming a single pair and the third from the two ways of forming two pairs. In the ﬁnal line we have put the OPE in standard form by Taylor expanding inside the normal ordering to express everything in terms of local operators at z and putting the most singular terms ﬁrst. Another important example is F = eik1 ·X(z,¯z ) ,

G = eik2 ·X(0,0) .

(2.2.12)

The variations δ/δXFµ and δ/δXGµ give factors of ik1µ and ik2µ respectively, so the general result (2.2.10) becomes α

: eik1 ·X(z,¯z ) : : eik2 ·X(0,0) : = exp = |z|

2

α k1 ·k2

k1 · k2 ln |z|2 : eik1 ·X(z,¯z ) eik2 ·X(0,0) : : eik1 ·X(z,¯z ) eik2 ·X(0,0) : .

(2.2.13)

To derive the OPE, Taylor expand inside the normal ordering to give

: eik1 ·X(z,¯z ) : : eik2 ·X(0,0) : = |z|α k1 ·k2 : ei(k1 +k2 )·X(0,0) [1 + O(z, ¯z )] : .

(2.2.14)

The exercises give further practice with normal ordering and the free-ﬁeld OPE. Note that the OPEs (2.2.2), (2.2.4), and so on have been written asymmetrically in σ1 and σ2 , expanding around the latter point. They can also be cast in symmetric form by Taylor expanding the right-hand sides around (σ1 + σ2 )/2. The coeﬃcient functions for the symmetric form behave simply under interchange of the two operators, ckij (σ1 − σ2 )sym = ±ckji (σ2 − σ1 )sym ,

(2.2.15)

where the minus sign appears if Ai and Aj are both anticommuting. The asymmetric form is usually more convenient for calculation, so when the

41

2.3 Ward identities and Noether’s theorem

symmetry properties of the coeﬃcient functions are needed one can work them out in the symmetric form and then convert to the asymmetric form. 2.3

Ward identities and Noether’s theorem

World-sheet symmetries of course play an important role in string theory. In this section we ﬁrst derive some general consequences of symmetry in ﬁeld theory. Consider a general ﬁeld theory with action S [φ] in d spacetime dimensions, with φα (σ) denoting general ﬁelds. Let there be a symmetry φα (σ) = φα (σ) + δφα (σ) ,

(2.3.1)

where δφα is proportional to an inﬁnitesimal parameter . The product of the path integral measure and the weight exp(−S ) is invariant, [dφ ] exp(−S [φ ]) = [dφ] exp(−S [φ]) .

(2.3.2)

A continuous symmetry in ﬁeld theory implies the existence of a conserved current (Noether’s theorem) and also Ward identities, which constrain the operator products of the current. To derive these results consider the change of variables, φα (σ) = φα (σ) + ρ(σ)δφα (σ) .

(2.3.3)

This is not a symmetry, the transformation law being altered by the inclusion of an arbitrary function ρ(σ). The path integral measure times exp(−S ) would be invariant if ρ were a constant, so its variation must be proportional to the gradient ∂a ρ, [dφ ] exp(−S [φ ])

= [dφ] exp(−S [φ]) 1 +

i 2π

dd σ g 1/2 j a (σ)∂a ρ(σ) + O(2 ) . (2.3.4)

The unknown coeﬃcient j a (σ) comes from the variation of the measure and the action, both of which are local, and so it must be a local function of the ﬁelds and their derivatives. Take the function ρ to be nonzero only in a small region, and consider a path integral with general insertions ‘. . .’ outside this region; the insertions are therefore invariant under (2.3.3). Invariance of the path integral under change of variables gives

0= =

[dφ ] exp(−S [φ ]) . . . −

2πi

[dφ] exp(−S [φ]) . . .

dd σ g 1/2 ρ(σ) ∇a j a (σ) . . . ,

(2.3.5)

where the limited support of ρ has allowed us to integrate by parts. Thus

42

2 Conformal ﬁeld theory

we have ∇a j a = 0

(2.3.6)

as an operator equation. This is Noether’s theorem, which is developed further in exercise 2.5. To derive the Ward identity, let ρ(σ) be 1 in some region R and 0 outside R. Also, include in the path integral some general local operator A(σ0 ) at a point σ0 inside R, and the usual arbitrary insertions ‘. . .’ outside. Proceeding as above we obtain the operator relation δA(σ0 ) + dd σ g 1/2 ∇a j a (σ)A(σ0 ) = 0 . (2.3.7) 2πi R Equivalently, ∇a j a (σ)A(σ0 ) = g −1/2 δ d (σ − σ0 )

2π δA(σ0 ) + total σ-derivative . (2.3.8) i

The divergence theorem gives

2π δA(σ0 ) (2.3.9) i ∂R with dA the area element and na the outward normal. This relates the integral of the current around the operator to the variation of the operator. Going to two ﬂat dimensions this becomes

dA na j a A(σ0 ) =

2π (2.3.10) δA(z0 , ¯z0 ) . ∂R Again we drop indices, j ≡ jz , ˜ ≡ j¯z ; notice that on a current the omitted indices are implicitly lower. We use a tilde rather than a bar on ˜ because this is not the adjoint of j. The Minkowski density j0 is in general Hermitean, so the Euclidean j2 with an extra factor of i is anti-Hermitean, and (jz )† = 12 (j1 − ij2 )† = jz . It is important that Noether’s theorem and the Ward identity are local properties, which do not depend on whatever boundary conditions we might have far away, nor even on whether these are invariant under the symmetry. In particular, since the function ρ(σ) is nonzero only inside R, the symmetry transformation need only be deﬁned there. In conformally invariant theories it is usually the case that jz is holomorphic and j¯z antiholomorphic (except for singularities at the other ﬁelds). In this case the currents (jz , 0) and (0, j¯z ) are separately conserved. The integral (2.3.10) then picks out the residues in the OPE, (jdz − ˜ d¯z )A(z0 , ¯z0 ) =

1 δA(z0 , ¯z0 ) . (2.3.11) i Here ‘Res’ and ‘Res’ are the coeﬃcients of (z − z0 )−1 and (¯z − ¯z0 )−1 respectively. This form of the Ward identity is particularly convenient. Resz→z0 j(z)A(z0 , ¯z0 ) + Res¯z →¯z0 ˜ (¯z )A(z0 , ¯z0 ) =

2.4 Conformal invariance

43

The world-sheet current was deﬁned with an extra factor of 2πi relative to the usual deﬁnition in ﬁeld theory in order to make this OPE simple. As an example, return to the free massless scalar and consider the spacetime translation δX µ = aµ . Under δX µ (σ) = ρ(σ)aµ , aµ δS = d2 σ ∂ a X µ ∂a ρ . (2.3.12) 2πα This is of the claimed form (2.3.5) with Noether current aµ jaµ , where i ∂a X µ . (2.3.13) α The components are holomorphic and antiholomorphic as expected. For the OPE of this current with the exponential operator one ﬁnds jaµ =

k µ ik·X(0,0) :e :, (2.3.14a) 2z k µ ik·X(0,0) ˜ µ (¯z ) : eik·X(0,0) : ∼ :, (2.3.14b) :e 2¯z from terms with a single contraction. This OPE is in agreement with the general identity (2.3.11). Another example is the world-sheet translation δσ a = v a , under which δX µ = −v a ∂a X µ . The Noether current is j µ (z) : eik·X(0,0) : ∼

Here Tab

ja = iv b Tab , 1 1 Tab = − : ∂a X µ ∂b Xµ − δab ∂c X µ ∂c Xµ : . α 2 is the world-sheet energy-momentum tensor.2 2.4

(2.3.15a) (2.3.15b)

Conformal invariance

The energy-momentum tensor (2.3.15b) is traceless, Taa = 0. In complex coordinates this is Tz¯z = 0 . The conservation

∂a Tab

(2.4.1)

= 0 then implies that in any theory with Taa = 0, ¯ zz = ∂T¯z¯z = 0 . ∂T (2.4.2)

Thus T (z) ≡ Tzz (z) , 2

˜ (¯z ) ≡ T¯z¯z (¯z ) T

(2.4.3)

In Tab we have used normal ordering to deﬁne the product of operators at a point. The only possible ambiguity introduced by the renormalization is a constant times δab , from the subtraction. Adding such a constant gives a diﬀerent energy-momentum tensor which is also conserved. We choose to focus on the tensor (2.3.15b), for reasons that will be explained more fully in the next chapter.

44

2 Conformal ﬁeld theory

are respectively holomorphic and antiholomorphic. For the free massless scalar, 1 ¯ µ: , ¯ µ ∂X ˜ (¯z ) = − 1 : ∂X T (z) = − : ∂X µ ∂Xµ : , T (2.4.4) α α which are indeed holomorphic and antiholomorphic as a consequence of the equation of motion. The tracelessness of Tab implies a much larger symmetry. The currents ˜ (¯z ) j(z) = iv(z)T (z) , ˜ (¯z ) = iv(z)∗ T (2.4.5) are conserved for any holomorphic v(z). For the free scalar theory, one ﬁnds the OPE 1 ¯ µ (0) . ˜ (¯z )X µ (0) ∼ 1 ∂X (2.4.6) T (z)X µ (0) ∼ ∂X µ (0) , T ¯z z The Ward identity then gives the transformation ¯ µ. (2.4.7) δX µ = −v(z)∂X µ − v(z)∗ ∂X This is an inﬁnitesimal coordinate transformation z = z + v(z). The ﬁnite transformation is X µ (z , ¯z ) = X µ (z, ¯z ) ,

z = f(z)

(2.4.8)

for any holomorphic f(z). This is known as a conformal transformation. The conformal symmetry we have found should not be confused with the diﬀ invariance of general relativity. We are in the ﬂat space theory, with no independent metric ﬁeld to vary, so the transformation z → z actually changes the distances between points. We would not ordinarily have such an invariance; it is a nontrivial statement about the dynamics. For the scalar action (2.1.10), the conformal transformation of ∂ and ∂¯ just balances that of d2 z. A mass term m2 X µ Xµ would not be invariant. Obviously there will in the end be a close relation with the diﬀ×Weyl symmetry of the Polyakov string, but we leave that for the next chapter. Consider the special case z = ζz

(2.4.9)

for complex ζ. The phase of ζ is a rotation of the system, while its magnitude is a rescaling of the size of the system. Such a scale invariance has occasionally been considered as an approximate symmetry in particle physics, and statistical systems at a critical point are described by scaleinvariant ﬁeld theories. To get some insight into the general conformal transformation, consider its eﬀect on inﬁnitesimal distances ds2 = dσ a dσ a = dzd¯z . Conformal transformations rescale this by a position-dependent factor, ds2 = dz d¯z =

∂z ∂¯z dzd¯z . ∂z ∂¯z

(2.4.10)

2.4 Conformal invariance

(a)

(b)

45

(c)

Fig. 2.2. (a) Two-dimensional region. (b) Eﬀect of the special conformal transformation (2.4.9). (c) Eﬀect of a more general conformal transformation.

Thus, as indicated in ﬁgure 2.2, a conformal transformation takes inﬁnitesimal squares into inﬁnitesimal squares, but rescales them by a positiondependent factor. An antiholomorphic function z = f(z)∗ has the same property, but changes the orientation. Most systems that are invariant under the rigid scaling (2.4.9) are actually invariant under the much larger conformal symmetry. A theory with this invariance is called a conformal ﬁeld theory (CFT). Conformal invariance in more than two dimensions is developed in exercise 2.6.

Conformal invariance and the OPE Conformal invariance puts strong constraints on the form of the OPE, and in particular on the OPEs of the energy-momentum tensor. Consider ˜ (¯z ) the OPE of T with the general operator A. Because T (z) and T are (anti)holomorphic except at insertions, the corresponding coeﬃcient functions must also have this property. The OPE of T with a general A is therefore a Laurent expansion, in integer but possibly negative powers of z. Further, all the singular terms are determined by the conformal transformation of A. To see this, let us write a general expansion of the singular terms, T (z)A(0, 0) ∼

∞ n=0

1 z n+1

A(n) (0, 0)

(2.4.11)

˜ ; the operator coeﬃcients A(n) remain to be determined. and similarly for T Under an inﬁnitesimal conformal transformation z = z + v(z), a single pole in v(z)T (z)A(0, 0) arises when the z −n−1 term of the T A OPE multiplies the term of order z n in v(z). Thus, the Ward identity in the

46

2 Conformal ﬁeld theory

form (2.3.11) implies that δA(z, ¯z ) = −

∞ 1 n ˜ (n) (z, ¯z ) . ∂ v(z)A(n) (z, ¯z ) + ∂¯n v(z)∗ A n=0

n!

(2.4.12)

Thus the operators A(n) are determined by the conformal transformation of A. It is convenient to take a basis of local operators that are eigenstates under rigid transformation (2.4.9), ˜

A (z , ¯z ) = ζ −h ¯ζ −h A(z, ¯z ) .

(2.4.13)

The (h, ˜h) are known as the weights of A. The sum h + h˜ is the dimension of A, determining its behavior under scaling, while h − ˜h is the spin, determining its behavior under rotations. The derivative ∂z increases h by one, and the derivative ∂¯z increases ˜h by one. The Ward identities for the transformation (2.4.13) and for the translation δA = −v a ∂a A determine part of the OPE, T (z)A(0, 0) = . . . +

1 h A(0, 0) + ∂A(0, 0) + . . . , 2 z z

(2.4.14)

˜. and similarly for T An important special case is a tensor operator or primary ﬁeld 3 O, on which a general conformal transformation acts as ˜

O (z , ¯z ) = (∂z z )−h (∂¯z ¯z )−h O(z, ¯z ) .

(2.4.15)

The OPE (2.4.11) reduces to T (z)O(0, 0) =

h 1 O(0, 0) + ∂O(0, 0) + . . . , 2 z z

(2.4.16)

the more singular terms in the general OPE (2.4.14) being absent. Taking again the example of the free X µ CFT, the weights of some typical operators are Xµ ¯ µ ∂X

(0, 0) , ∂X µ (1, 0) , (0, 1) , ∂2 X µ (2, 0) , 2 2 αk αk : eik·X : . , 4 4

3

(2.4.17)

In quantum ﬁeld theory, one usually distinguishes the fundamental ﬁelds (the variables of integration in the path integral) from more general operators which may be composite. Actually, this distinction is primarily useful in weakly coupled ﬁeld theories in three or more dimensions. It is of little use in CFT, in particular because of equivalences between diﬀerent ﬁeld theories, and so the term ﬁeld is used for any local operator.

47

2.4 Conformal invariance

All transform as tensors except ∂2 X µ . More generally, an exponential times a general product of derivatives, :

i∂

m i X µi

j

∂¯nj X νj eik·X : ,

(2.4.18)

has weight 2 αk

4

+

α k 2 + i mi , 4

j

nj

.

(2.4.19)

For any pair of operators, applying rigid translations, scale transformations, and rotations to both sides of the OPE determines the z-dependence of the coeﬃcient functions completely, Ai (z1 , ¯z1 )Aj (z2 , ¯z2 ) =

hk −hi −hj ˜hk −˜hi −˜hj ¯z12 z12 ckij Ak (z2 , ¯z2 ) ,

(2.4.20)

k

where the ckij are now constants. In all cases of interest the weights appearing on the right-hand side of the OPE (2.4.20) are bounded below, and so the degree of singularity in the operator product is bounded. More general conformal transformations put further constraints on the OPE: they determine the OPEs of all ﬁelds in terms of those of the primary ﬁelds. We will develop this in chapter 15. Notice that the conformal transformation properties of normal ordered products are not in general given by the naive transformations of the product. For example, the transformation law (2.4.7) for X µ would naively imply ¯ ik·X (naive) (2.4.21) δeik·X = −v(z)∂eik·X − v(z)∗ ∂e for the exponential, making it a tensor of weight (0, 0). The modiﬁcation is a quantum eﬀect, due to the renormalization needed to deﬁne the product of operators at a point. Speciﬁcally it enters here because the subtraction ln |z12 |2 in : : makes explicit reference to the coordinate frame. Conformal properties of the energy-momentum tensor The OPE of the energy-momentum tensor with itself was obtained in eq. (2.2.11), η µµ 2 − : ∂X µ (z)∂Xµ (0) : + : T (z)T (0) : 2z 4 α z 2 D 2 1 ∼ 4 + 2 T (0) + ∂T (0) . (2.4.22) 2z z z ˜ . The T (z)T ˜ (¯z ) OPE must be nonsingular. A similar result holds for T It cannot have poles in (z − z ) because it is antiholomorphic in z at nonzero separation; similarly it cannot have poles in (¯z − ¯z ) because it is T (z)T (0) =

48

2 Conformal ﬁeld theory

holomorphic in z at nonzero separation.4 The same holds for any OPE between a holomorphic and an antiholomorphic operator. Thus T is not a tensor. Rather, the OPE (2.4.22) implies the transformation law D (2.4.23) −1 δT (z) = − ∂z3 v(z) − 2∂z v(z)T (z) − v(z)∂z T (z) . 12 In a general CFT, T (z) transforms as c −1 δT (z) = − ∂z3 v(z) − 2∂z v(z)T (z) − v(z)∂z T (z) , (2.4.24) 12 with c a constant known as the central charge. The central charge of a free scalar is 1; for D free scalars it is D. The transformation (2.4.24) is the most general form that is linear in v, is consistent with the symmetry of the T T OPE, and has three lower z indices as required by rigid scale and rotation invariance. The scale, rotation, and translation symmetries determine the coeﬃcients of the second and third terms. Further, by considering the commutator of two such transformations one can show that ∂a c = 0, and it is a general result in quantum ﬁeld theory that an operator that is independent of position must be a c-number.5 The corresponding T T OPE is 2 1 c T (z)T (0) ∼ 4 + 2 T (0) + ∂T (0) . (2.4.25) 2z z z The ﬁnite form of the transformation law (2.4.24) is c (2.4.26) (∂z z )2 T (z ) = T (z) − {z , z} , 12 where {f, z} denotes the Schwarzian derivative {f, z} =

2∂z3 f∂z f − 3∂z2 f∂z2 f . 2∂z f∂z f

(2.4.27)

One can check this by verifying that it has the correct inﬁnitesimal form and that it composes correctly under successive transformations (so that one can integrate the inﬁnitesimal transformation). The corresponding ˜ , possibly with a diﬀerent central charge ˜c in the general forms hold for T CFT. The nontensor behavior of the energy-momentum tensor has a number of important physical consequences, as we will see. We should empha4

5

Unless otherwise stated OPEs hold only at nonzero separation, ignoring possible delta functions. For our applications the latter will not matter. Occasionally it is useful to include the delta functions, but in general these depend partly on deﬁnitions so one must be careful. The argument is this. Suppose that A(σ) is independent of position, and consider its equal time commutator with any other local operator B(σ ). These commute at spacelike separation, by locality. Since A(σ) is actually independent of position it also commutes at zero separation. Therefore A(σ) commutes with all local operators, and so must be a c-number.

2.5 Free CFTs

49

size that ‘nontensor’ refers to conformal transformations. The energymomentum tensor will have its usual tensor property under coordinate transformations. 2.5

Free CFTs

In this section we discuss three families of free-ﬁeld CFTs — the linear dilaton, bc, and βγ theories. The bc theory is the one of most immediate interest, as it will appear in the next chapter when we gauge-ﬁx the Polyakov string, but all have a variety of applications in string theory. Linear dilaton CFT This family of CFTs is based on the same action (2.1.10) but with energymomentum tensor 1 (2.5.1a) T (z) = − : ∂X µ ∂Xµ : +Vµ ∂2 X µ , α ¯ µ : +Vµ ∂¯2 X µ , ¯ µ ∂X ˜ (¯z ) = − 1 : ∂X T (2.5.1b) α where Vµ is some ﬁxed D-vector. Working out the T T OPE, one ﬁnds that it is of the standard form (2.4.25), but with central charge c = ˜c = D + 6α Vµ V µ .

(2.5.2)

The T X µ OPE and the Ward identity (2.4.12) imply the conformal transformation ¯ µ − α V µ [∂v + (∂v)∗ ] . (2.5.3) δX µ = −v∂X µ − v ∗ ∂X 2 This is a diﬀerent conformal symmetry of the same action. The ﬁeld X µ no longer transforms as a tensor, its variation now having an inhomogeneous piece. Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry — much more than we will have occasion to mention. The energy-momentum tensor plays a special role in string theory — in particular, it tells us how to couple to a curved metric — so diﬀerent values of V µ are to be regarded as diﬀerent CFTs. The vector V µ picks out a direction in spacetime. This CFT is therefore not Lorentz-invariant and not of immediate interest to us. We will see one physical interpretation of this linear dilaton CFT in section 3.7, and encounter it in some technical applications later. A diﬀerent variation of the free scalar CFT is to take some of the X µ to be periodic; we will take this up in chapter 8.

50

2 Conformal ﬁeld theory bc CFT

The second family of CFTs has anticommuting ﬁelds b and c with action 1 ¯ . S= d2 z b∂c (2.5.4) 2π This is conformally invariant for b and c transforming as tensors of weights (hb , 0) and (hc , 0) such that hb = λ ,

hc = 1 − λ

(2.5.5)

for any given constant λ. Thus we have another family of CFTs (which is secretly the same as the linear dilaton family, as we will learn in chapter 10). The operator equations of motion, obtained by the same method (2.1.15), (2.1.18) as before, are ¯ ¯ ∂c(z) = ∂b(z) =0, (2.5.6a) 2 ¯ (2.5.6b) ∂b(z)c(0) = 2πδ (z, ¯z ) . The bb and cc OPEs satisfy the equation of motion without source. The normal ordered bc product is 1 : b(z1 )c(z2 ) : = b(z1 )c(z2 ) − . (2.5.7) z12 This satisﬁes the naive equations of motion as a consequence of 1 1 ∂¯ = ∂ = 2πδ 2 (z, ¯z ) , (2.5.8) ¯z z which can be veriﬁed by integrating over a region containing the origin and integrating the derivative by parts. Normal ordering of a general product of ﬁelds is combinatorially the same as for the X µ CFT, a sum over contractions or subtractions. One must be careful because b and c are anticommuting so that interchange of ﬁelds ﬂips the sign: one should anticommute the ﬁelds being paired until they are next to each other before replacing them with the subtraction (2.5.7). The operator products are 1 1 , c(z1 )b(z2 ) ∼ , (2.5.9) b(z1 )c(z2 ) ∼ z12 z12 where in the second OPE there have been two sign ﬂips, one from anticommutation and one from z1 ↔ z2 . Other operator products are nonsingular: b(z1 )b(z2 ) = O(z12 ) ,

c(z1 )c(z2 ) = O(z12 ) .

(2.5.10)

These are not only holomorphic but have a zero due to antisymmetry. Noether’s theorem gives the energy-momentum tensor T (z) = : (∂b)c : −λ∂(: bc :) , ˜ (¯z ) = 0 . T

(2.5.11a) (2.5.11b)

51

2.5 Free CFTs

One can also verify (2.5.11) by working out the OPE of T with b and c; it has the standard tensor form (2.4.16) with the given weights. The T T OPE is of standard form (2.4.25) with c = −3(2λ − 1)2 + 1 ,

˜c = 0 .

(2.5.12)

This is a purely holomorphic CFT, and is an example where c = ˜c. There is of course a corresponding antiholomorphic theory

1 d2 z ˜b∂˜c , (2.5.13) 2π which is the same as the above with z ↔ ¯z . The bc theory has a ghost number symmetry δb = −ib, δc = ic. The corresponding Noether current is S=

j = − : bc : .

(2.5.14)

Again the components are separately holomorphic and antiholomorphic, the latter vanishing. When there are both holomorphic and antiholomorphic bc ﬁelds, the ghost numbers are separately conserved. This current is not a tensor, 1 1 1 − 2λ + 2 j(0) + ∂j(0) . z3 z z This implies the transformation law T (z)j(0) ∼

(2.5.15)

2λ − 1 2 ∂ v, 2

(2.5.16)

2λ − 1 ∂z2 z . 2 ∂z z

(2.5.17)

−1 δj = −v∂j − j∂v + whose ﬁnite form is (∂z z )jz (z ) = jz (z) +

The one case where b and c have equal weight is hb = hc = 12 , for which the central charge c = 1. Here we will often use the notation b → ψ, c → ψ. For this case the bc CFT can be split in two in a conformally invariant way, ψ = 2−1/2 (ψ1 + iψ2 ) , ψ = 2−1/2 (ψ1 − iψ2 ) , (2.5.18a) 1 ¯ 1 + ψ2 ∂ψ ¯ 2 , S = d2 z ψ1 ∂ψ (2.5.18b) 4π 1 1 T = − ψ1 ∂ψ1 − ψ2 ∂ψ2 . (2.5.18c) 2 2 Each ψ theory has central charge 12 . The bc theory for λ = 2, weights (hb , hc ) = (2, −1), will arise in the next chapter as the Faddeev–Popov ghosts from gauge-ﬁxing the Polyakov string. The ψ theory will appear extensively in the more general string theories of volume two.

52

2 Conformal ﬁeld theory βγ CFT

The third family of CFTs is much like the bc theory but with commuting ﬁelds; β is an (hβ , 0) tensor and γ an (hγ , 0) tensor, where hβ = λ , The action is

hγ = 1 − λ .

(2.5.19)

1 ¯ . d2 z β ∂γ 2π These ﬁelds are again holomorphic by the equations of motion, ¯ ¯ ∂γ(z) = ∂β(z) =0. S=

(2.5.20)

(2.5.21)

The equations of motion and operator products are derived in the standard way. Because the statistics are changed, some signs in operator products are diﬀerent, 1 1 , γ(z1 )β(z2 ) ∼ . (2.5.22) β(z1 )γ(z2 ) ∼ − z12 z12 The energy-momentum tensor is T = : (∂β)γ : −λ∂(: βγ :) , ˜ =0. T

(2.5.23a) (2.5.23b)

The central charge is simply reversed in sign, c = 3(2λ − 1)2 − 1 ,

˜c = 0 .

(2.5.24)

The βγ theory for λ = 32 , weights (hβ , hγ ) = ( 32 , − 12 ), will arise in chapter 10 as the Faddeev–Popov ghosts from gauge-ﬁxing the superstring. 2.6

The Virasoro algebra

Thus far in this chapter we have studied local properties of the twodimensional ﬁeld theory. We now are interested in the spectrum of the theory. The spatial coordinate will be periodic, as in the closed string, or bounded, as in the open string. For the periodic case let σ 1 ∼ σ 1 + 2π .

(2.6.1)

Let the Euclidean time coordinate run − ∞ < σ2 < ∞

(2.6.2)

so that the two dimensions form an inﬁnite cylinder. It is again useful to form a complex coordinate, and there are two natural choices. The ﬁrst is w = σ 1 + iσ 2 ,

(2.6.3)

2.6 The Virasoro algebra

w

53

z

2π

(a) 0

(b)

Fig. 2.3. Closed string coordinates. (a) Equal time contours in the w-plane. The dashed lines are identiﬁed. (b) The same contours in the z-plane.

so that w ∼ w + 2π. The second is z = exp(−iw) = exp(−iσ 1 + σ 2 ) .

(2.6.4)

These two coordinate systems are shown in ﬁgure 2.3. In terms of the w coordinate, time corresponds to translation of σ 2 = Im w. In terms of z, time runs radially, the origin being the inﬁnite past. These coordinates are related by a conformal transformation. The w coordinate is natural for the canonical interpretation of the theory, but the z coordinate is also quite useful and most expressions are written in this frame. For a holomorphic or antiholomorphic operator we can make a Laurent expansion, Tzz (z) =

∞

Lm , m+2 m=−∞ z

˜¯z¯z (¯z ) = T

∞

˜m L . z m+2 m=−∞ ¯

(2.6.5)

The Laurent coeﬃcients, known as the Virasoro generators, are given by the contour integrals

Lm =

C

dz m+2 Tzz (z) , z 2πiz

(2.6.6)

where C is any contour encircling the origin counterclockwise. The Laurent expansion is just the same as an ordinary Fourier transformation in the w frame at time σ 2 = 0: Tww (w) = −

∞

exp(imσ 1 − mσ 2 )Tm ,

m=−∞ ∞

¯ =− Tw¯ w¯ (w)

˜m , exp(−imσ 1 − mσ 2 )T

m=−∞

(2.6.7a) (2.6.7b)

54

2 Conformal ﬁeld theory

where c ˜m = L ˜ m − δm,0 ˜c . , T (2.6.8) 24 24 The additive shift of T0 is from the nontensor transformation (2.4.26), c Tww = (∂w z)2 Tzz + . (2.6.9) 24 Tm = Lm − δm,0

The Hamiltonian H of time translation in the w = σ 1 + iσ 2 frame is H=

2π dσ 1 0

˜ 0 − c + ˜c . T22 = L0 + L 2π 24

(2.6.10)

Notice that the +2s from the Laurent expansions (2.6.5) have canceled in the Fourier expansions (2.6.7) due to the conformal transformation of T . Similarly the Laurent expansion for a holomorphic ﬁeld of weight h would include +h in the exponent. Cutting open the path integral on circles of constant time Im w = ln |z|, the Virasoro generators become operators in the ordinary sense. (This idea of cutting open a path integral is developed in the appendix.) By holomorphicity the integrals (2.6.6) are independent of C and so in particular are invariant under time translation (radial rescaling). That is, they are conserved charges, the charges associated with the conformal transformations. It is an important fact that the OPE of currents determines the algebra of the corresponding charges. Consider general charges Qi , i = 1, 2, given as contour integrals of holomorphic currents, Qi {C} =

C

dz ji . 2πi

(2.6.11)

Consider the combination Q1 {C1 }Q2 {C2 } − Q1 {C3 }Q2 {C2 } ,

(2.6.12)

where the contours are shown in ﬁgure 2.4(a). The order in which the factors are written is irrelevant, as these are just variables of integration in a path integral (unless both charges are anticommuting, in which case there is an additional sign and all the commutators become anticommutators). As discussed in the appendix, when we slice open the path integral to make an operator interpretation, what determines the operator ordering is the time ordering, which here is t1 > t2 > t3 . The path integral with the combination (2.6.12) thus corresponds to a matrix element of ˆ 1Q ˆ 2Q ˆ 1, Q ˆ 2] . ˆ2 −Q ˆ 1 ≡ [Q Q

(2.6.13)

Now, for a given point z2 on the contour C2 , we can deform the diﬀerence of the C1 and C3 contours as shown in ﬁgure 2.4(b), so the commutator

55

2.6 The Virasoro algebra

z=0 C3

C2

C1

C2 z2 C1 C3

(a)

(b)

Fig. 2.4. (a) Contours centered on z = 0. (b) For given z2 on contour C2 , contour C1 − C3 is contracted.

is given by the residue of the OPE, [Q1 , Q2 ]{C2 } =

C2

dz2 Resz1 →z2 j1 (z1 )j2 (z2 ) . 2πi

(2.6.14)

This contour argument allows us to pass back and forth between OPEs and commutation relations. Let us emphasize this: for conserved currents, knowing the singular terms in the OPE is equivalent to knowing the commutator algebra of the corresponding charges. The calculation of ﬁgure 2.4 also applies with the conserved charge Q2 {C2 } replaced by any operator, 1 (2.6.15) δA(z2 , ¯z2 ) . i This is just the familiar statement that a charge Q generates the corresponding transformation δ. Similarly for the contour integral of an antiholomorphic current [Q, A(z2 , ¯z2 )] = Resz1 →z2 j(z1 )A(z2 , ¯z2 ) =

d¯z ˜ , 2πi C the Ward identity and contour argument imply ˜ Q{C} =−

(2.6.16)

1 (2.6.17) δA(z2 , ¯z2 ) . i Apply this to the Virasoro generators (2.6.6), where jm (z) = z m+1 T (z): ˜ A(z2 , ¯z2 )] = Res¯z →¯z ˜ (¯z1 )A(z2 , ¯z2 ) = [Q, 1 2

Resz1 →z2 z1m+1 T (z1 ) z2n+1 T (z2 )

=

Resz1 →z2 z1m+1 z2n+1

c 2 1 + 2 T (z2 ) + ∂T (z2 ) 4 z12 2z12 z12

56

2 Conformal ﬁeld theory c 3 m+1 n+1 (∂ z2 )z2 + 2(∂z2m+1 )z2n+1 T (z2 ) + z2m+n+2 ∂T (z2 ) 12 c 3 = (m − m)z2m+n−1 + (m − n)z2m+n+1 T (z2 ) + total derivative . 12 (2.6.18)

=

The z2 contour integral of the right-hand side then gives the Virasoro algebra, c [Lm , Ln ] = (m − n)Lm+n + (m3 − m)δm,−n . (2.6.19) 12 ˜ m satisfy the same algebra with central charge ˜c. The L Any CFT thus has an inﬁnite set of conserved charges, the Virasoro generators, which act in the Hilbert space and which satisfy the algebra (2.6.19). Let us for now notice just a few simple properties. Generally ˜ 0 . The generator L0 satisﬁes we work with eigenstates of L0 and L [L0 , Ln ] = −nLn .

(2.6.20)

If |ψ is an eigenstate of L0 with eigenvalue h, then L0 Ln |ψ = Ln (L0 − n)|ψ = (h − n)Ln |ψ ,

(2.6.21)

so that Ln |ψ is an eigenstate with eigenvalue h − n. The generators with n < 0 raise the L0 eigenvalue and those with n > 0 lower it. The three generators L0 and L±1 form a closed algebra without central charge, [L0 , L1 ] = −L1 ,

[L0 , L−1 ] = L−1 ,

[L1 , L−1 ] = 2L0 .

(2.6.22)

This is the algebra S L(2, R), which diﬀers from S U(2) by signs. For the Laurent coeﬃcients of a holomorphic tensor ﬁeld O of weight (h, 0), O(z) =

∞

Om , m+h z m=−∞

(2.6.23)

one ﬁnds from the OPE (2.4.16) the commutator [Lm , On ] = [(h − 1)m − n]Om+n .

(2.6.24)

Again modes with n > 0 reduce L0 , while modes with n < 0 increase it. In the open string, let 0 ≤ Re w ≤ π

⇔

Im z ≥ 0 ,

(2.6.25)

where z = − exp(−iw). These coordinate regions are shown in ﬁgure 2.5. At a boundary, the energy-momentum tensor satisﬁes Tab na tb = 0 ,

(2.6.26)

where na and tb are again normal and tangent vectors. To see this, consider a coordinate system in which the boundary is straight. The presence of

2.6 The Virasoro algebra

57

w z

(a) 0

π

(b)

Fig. 2.5. Open string coordinates. (a) Equal time contours in the w-plane. (b) The same contours in the z-plane. The dashed line shows the extension of one contour, as used in the doubling trick.

the boundary breaks translation invariance in the normal direction but not the tangential, so that the current Tab tb is still conserved. Then the boundary condition (2.6.26) is just the statement that the ﬂow of this current out of the boundary is zero. In the present case, this becomes Tww = Tw¯ w¯ , Re w = 0, π

⇔

Tzz = T¯z¯z , Im z = 0 .

(2.6.27)

It is convenient to use the doubling trick. Deﬁne Tzz in the lower halfz-plane as the value of T¯z¯z at its image in the upper half-z-plane, z = ¯z : Tzz (z) ≡ T¯z¯z (¯z ) ,

Imz < 0 .

(2.6.28)

The equation of motion and boundary condition are then summarized by the statement that Tzz is holomorphic in the whole complex plane. There is only one set of Virasoro generators, because the boundary condition ˜, couples T and T

1 dz z m+1 Tzz − d¯z ¯z m+1 T¯z¯z Lm = 2πi C 1 = dz z m+1 Tzz (z) . 2πi

(2.6.29)

In the ﬁrst line, the contour C is a semi-circle centered on the origin; in the second line, we have used the doubling trick to write Lm in terms of a closed contour. Again, these satisfy the Virasoro algebra [Lm , Ln ] = (m − n)Lm+n +

c 3 (m − m)δm,−n . 12

(2.6.30)

58

2 Conformal ﬁeld theory 2.7

Mode expansions Free scalars

In free ﬁeld theory, the ﬁelds break up into harmonic oscillators, and the spectrum and energy-momentum tensor can be given in terms of the ¯ are modes. We start with the closed string. In the X µ theory, ∂X and ∂X (anti)holomorphic and so have Laurent expansions like that for T , ∂X µ (z) = −i

1/2 ∞ α

2

αµm

m=−∞

z m+1

¯ µ (¯z ) = −i α ∂X 2

,

1/2 ∞

˜αµm . z m+1 m=−∞ ¯ (2.7.1)

Equivalently,

2 α

1/2

dz m µ z ∂X (z) , 2π 1/2 2 d¯z m ¯ µ ¯z ∂X (¯z ) . α˜µm = − α 2π αµm =

(2.7.2a) (2.7.2b)

Single-valuedness of X µ implies that αµ0 = α˜µ0 . Moreover, the Noether current for spacetime translations is i∂a X µ /α , so the spacetime momentum is 1 p = 2πi

µ

(dz j − d¯z ˜ ) = µ

C

µ

2 α

1/2

αµ0

=

2 α

1/2

˜αµ0 .

(2.7.3)

Integrating the expansions (2.7.1) gives

α α X (z, ¯z ) = x − i pµ ln |z|2 + i 2 2 µ

µ

1/2 ∞ m=−∞ m=0

1 αµm α˜µm + m z m ¯z m

.

(2.7.4)

Either from standard canonical commutation, or from the contour argument and the XX OPE, one derives [αµm , ανn ] = [˜αµm , ˜ανn ] = mδm,−n η µν ,

(2.7.5a)

[xµ , pν ] = iη µν ,

(2.7.5b)

with other commutators vanishing. The spectrum is given by starting with a state |0; k that has momentum k µ and is annihilated by all of the lowering modes, αµn for n > 0, and acting in all possible ways with the raising (n < 0) modes. We now wish to expand the Virasoro generators in terms of the mode operators. Insert the Laurent expansion for X µ into the energy-momentum

2.7 Mode expansions

59

tensor (2.4.4) and collect terms with a given power of z, giving Lm ∼

∞ 1 αµm−n αµn . 2 n=−∞

(2.7.6)

The ∼ indicates that we have ignored the ordering of operators. For m = 0, the expansion (2.7.6) is well deﬁned and correct as it stands — the mode operators in each term commute and so the ordering does not matter. For m = 0, put the lowering operators on the right and introduce a normal ordering constant, L0 =

∞ α p2 + (αµ−n αµn ) + aX . 4 n=1

(2.7.7)

We encountered this same issue in section 1.3, where we treated it in a heuristic way. Now the left-hand side has a ﬁnite and unambiguous deﬁnition in terms of the Laurent coeﬃcients of the : :-ordered energymomentum tensor, so the normal ordering constant is ﬁnite and calculable. There are several ways to determine it. The simplest uses the Virasoro algebra, 2L0 |0; 0 = (L1 L−1 − L−1 L1 )|0; 0 = 0 ,

(2.7.8)

aX = 0 .

(2.7.9)

and so Here we have used the known form of L1 and L−1 : every term of each contains either a lowering operator or pµ and so annihilates |0; 0 . We determined the central charge in the Virasoro algebra from the OPE. It can also be determined directly from the expression for the Virasoro generators in terms of the mode operators, though some care is needed. This is left as exercise 2.11. Let us introduce a new notation. The symbol ◦◦ ◦◦ will denote creation– annihilation normal ordering, placing all lowering operators to the right of all raising operators, with a minus sign whenever anticommuting operators are switched. For the purposes of this deﬁnition, we will include pµ with the lowering operators and xµ with the raising operators. In this notation we can write Lm =

∞ 1 ◦ µ α ◦◦ ◦α 2 n=−∞ m−n µn

(2.7.10)

since aX = 0. We have now introduced two forms of normal ordering, conformal normal ordering : : (which is what we will mean if we just refer to ‘normal ordering’) and creation–annihilation normal ordering ◦◦ ◦◦ . The former is useful because it produces operators whose OPEs and conformal

60

2 Conformal ﬁeld theory

transformation properties are simple. The latter, which is probably more familiar to the reader, is useful for working out the matrix elements of the operators. Let us work out the relation between them. We start by comparing the time-ordered and creation–annihilation-ordered products. For the product X µ (z, ¯z )X ν (z , ¯z ) with |z| > |z |, insert the mode expansions and move the lowering operators in X µ (z, ¯z ) to the right. Keeping track of the commutators gives X µ (z, ¯z )X ν (z , ¯z ) = ◦◦ X µ (z, ¯z )X ν (z , ¯z )◦◦

∞ 1 z m ¯z m α + η µν − ln |z|2 + + m ¯z 2 m zm m=1

= ◦◦ X µ (z, ¯z )X ν (z , ¯z )◦◦ −

α µν η ln |z − z |2 . 2

(2.7.11)

Since |z| > |z |, the left-hand side here is time-ordered and the sum converges. The deﬁnition (2.1.21) gave the relation between the timeordered product and the conformal-normal-ordered product (eq. (2.1.21) was in path integral language, so the product on its right becomes timeordered in operator formalism). It is the same as the relation (2.7.11), so that ◦ ◦

X µ (z, ¯z )X ν (z , ¯z )◦◦ = : X µ (z, ¯z )X ν (z , ¯z ) : .

(2.7.12)

Such a relation does not hold in general CFTs, and in fact the somewhat arbitrary grouping of pµ with the lowering operators was done in part to give this simple result. It also does not hold for operators conformalnormal-ordered in terms of w rather than z, for example. From eq. (2.7.12) one can write the mode expansion (2.7.10) at once, giving a second derivation of aX = 0. Creation–annihilation normal-ordered products of more than two X µ s have the same combinatoric properties as for : : ordering. That is, they are obtained from the time-ordered product by summing over all subtractions — this is Wick’s theorem from ﬁeld theory. Also, to convert operators normal-ordered in one form into a diﬀerent normal ordering, one sums over all subtractions using the diﬀerence of the two-point functions. That is, if we have two kinds of ordering, [X µ (z, ¯z )X ν (z , ¯z )]1 = [X µ (z, ¯z )X ν (z , ¯z )]2 + η µν ∆(z, ¯z , z , ¯z ) , then for a general operator F,

1 [F]1 = exp 2

(2.7.13)

δ δ d z d z ∆(z, ¯z , z , ¯z ) µ [F]2 . δX (z, ¯z ) δXµ (z , ¯z ) (2.7.14) For the linear dilaton CFT, the Laurent expansion and commutators 2

2

61

2.7 Mode expansions are unchanged, while the Virasoro generators contain the extra term

Lm =

∞ α 1 ◦ µ ◦ ◦ αm−n αµn ◦ + i 2 n=−∞ 2

1/2

(m + 1)V µ αµm .

(2.7.15)

bc CFT The ﬁelds b and c have the Laurent expansions ∞

bm

m=−∞

z m+λ

b(z) =

,

∞

c(z) =

m=−∞

cm m+1−λ z

.

(2.7.16)

To be precise, these are only Laurent expansions if λ is an integer, which we will assume for now. The half-integer case is also of interest, but will be dealt with in detail in chapter 10. The OPE gives the anticommutators {bm , cn } = δm,−n .

(2.7.17)

Consider ﬁrst the states that are annihilated by all of the n > 0 operators. The b0 , c0 oscillator algebra generates two such ground states | ↓ and | ↑ , with the properties b0 | ↓ = 0 , b0 | ↑ = | ↓ , c0 | ↓ = | ↑ , c0 | ↑ = 0 , bn | ↓ = bn | ↑ = cn | ↓ = cn | ↑ = 0 ,

n>0.

(2.7.18a) (2.7.18b) (2.7.18c)

The general state is obtained by acting on these states with the n < 0 modes at most once each (because these anticommute). For reasons to appear later it is conventional to group b0 with the lowering operators and c0 with the raising operators, so we will single out | ↓ as the ghost vacuum |0 . In string theory we will have a holomorphic bc theory and an antiholomorphic ˜b˜c theory, each with λ = 2. The closed string spectrum thus includes a product of two copies of the above. The Virasoro generators are ∞

Lm =

(mλ − n)◦◦ bn cm−n ◦◦ + δm,0 ag .

(2.7.19)

n=−∞

The ordering constant can be determined as in eq. (2.7.8), which gives 2L0 | ↓ = (L1 L−1 − L−1 L1 )| ↓

= (λb0 c1 )[(1 − λ)b−1 c0 ]| ↓ = λ(1 − λ)| ↓ .

(2.7.20)

Thus, ag = 12 λ(1 − λ) and Lm =

∞

(mλ − n)◦◦ bn cm−n ◦◦ +

n=−∞

λ(1 − λ) δm,0 . 2

(2.7.21)

62

2 Conformal ﬁeld theory

The constant can also be obtained by working out the relation between : : and ◦◦ ◦◦ for the CFT (exercises 2.13, 2.14). For the ghost number current (2.5.14), j = − : bc :, the charge is Ng = − =

1 2πi

∞

2π

dw jw

0

(c−n bn − b−n cn ) + c0 b0 −

n=1

1 . 2

(2.7.22)

It satisﬁes [N g , bm ] = −bm ,

[N g , cm ] = cm ,

(2.7.23)

and so counts the number of c minus the number of b excitations. The ground states have ghost number ± 12 : 1 1 N g | ↓ = − | ↓ , N g | ↑ = | ↑ . (2.7.24) 2 2 This depends on the value of the ordering constant, determined in exercise 2.13, but one might guess it on the grounds that the average ghost number of the ground states should be zero: the ghost number changes sign under b ↔ c. The βγ theory is similar; we leave the details until we need them in chapter 10. Open strings In the open string, the Neumann boundary condition becomes ∂z X µ = ∂¯z X µ on the real axis. There is only one set of modes, the boundary condition requiring αµm = ˜αµm in the expansions (2.7.1). The spacetime momentum integral (2.7.3) runs only over a semi-circle, so the normalization is now αµ0 = (2α )1/2 pµ .

(2.7.25)

The expansion for X µ is then µ

X (z, ¯z ) = x − iα p ln |z| + i µ

µ

2

1/2 ∞ α αµm

2

m=−∞ m=0

m

(z −m + ¯z −m ) .

(2.7.26)

Also, 2

L0 = α p +

∞ µ

α−n αµn .

(2.7.27)

[xµ , pν ] = iη µν .

(2.7.28)

n=1

The commutators are as before, [αµm , ανn ] = mδm,−n η µν ,

63

2.8 Vertex operators

For the bc theory, the boundary conditions that will be relevant to the string are c(z) = ˜c(¯z ) ,

b(z) = ˜b(¯z ) ,

Im z = 0 ,

(2.7.29)

written in terms of the z-coordinate where the boundary is the real axis. We can then use the doubling trick to write the holomorphic and antiholomorphic ﬁelds in the upper half-plane in terms of holomorphic ﬁelds in the whole plane, c(z) ≡ ˜c(¯z ) ,

b(z) ≡ ˜b(¯z ) ,

Im(z) ≤ 0 , z = ¯z .

(2.7.30)

The open string thus has a single set of Laurent modes for each of b and c. 2.8

Vertex operators

In quantum ﬁeld theory, there is on the one hand the space of states of the theory and on the other hand the set of local operators. In conformal ﬁeld theory there is a simple and useful isomorphism between these, with the CFT quantized on a circle. Consider the semi-inﬁnite cylinder in the w-coordinate, 0 ≤ Re w ≤ 2π ,

w ∼ w + 2π ,

Im w ≤ 0 ,

(2.8.1)

which maps into the unit disk in the coordinate z = exp(−iw). This is shown in ﬁgure 2.6. To deﬁne the path integral in the w-coordinate one must in particular specify the boundary condition as Im w → −∞. That is, one must specify the initial state. In the z-coordinate, Im w = −∞ maps to the origin, so this is equivalent to specifying the behavior of the ﬁelds at that point. In eﬀect, this deﬁnes a local operator at the origin, known as the vertex operator associated with the state. Going in the other direction, the path integral on the disk with an operator A at the origin maps to the path integral on the cylinder with a speciﬁed initial state |A . For free ﬁeld theories one can easily work out the detailed form of this isomorphism. Suppose one has a conserved charge Q acting on the state |A as in ﬁgure 2.6(a). One can ﬁnd the corresponding local operator by using the OPE to evaluate the contour integral in ﬁgure 2.6(b). Let us use this to identify the state |1 corresponding to the unit operator. With no ¯ µ are (anti)holomorphic inside the Q operator at the origin, ∂X µ and ∂X contour in ﬁgure 2.6(b). The contour integrals (2.7.2) deﬁning αµm and ˜αµm for m ≥ 0 then have no poles and so vanish. Thus |1 is annihilated by these modes. This identiﬁes it as the ground state, |1 = |0; 0 ,

(2.8.2)

64

2 Conformal ﬁeld theory

Q A

(a)

Q

(b)

A Fig. 2.6. (a) A semi-inﬁnite cylinder in w, with initial state |A and charge Q. (b) The conformally equivalent unit disk with local operator A and contour integral for Q.

with the normalization chosen for convenience. Now consider, for example, the state αµ−m |1 with m positive. Passing to ﬁgure 2.6(b) with Q = αµ−m surrounding the unit operator, the ﬁelds are holomorphic inside the contour, and so we can evaluate αµ−m =

2 α

1/2

dz −m µ z ∂X (z) → 2π

for m ≥ 1. Thus αµ−m |1 ∼ =

2 α

2 α

1/2

i ∂m X µ (0) (m − 1)!

1/2

i ∂m X µ (0) , (m − 1)!

(2.8.3)

m≥1.

(2.8.4)

m≥1.

(2.8.5)

Similarly ˜αµ−m |1

∼ =

2 α

1/2

i ∂¯m X µ (0) , (m − 1)!

This correspondence continues to hold when αµ−m or ˜αµ−m acts on a general state |A . If : A(0, 0) : is any normal-ordered operator, there may be singularities in the operator product of ∂X µ (z) with : A(0, 0) :, but it is not hard to check that αµ−m : A(0, 0) : = : αµ−m A(0, 0) :

(2.8.6)

65

2.8 Vertex operators

for m > 0 because the contour integral of the contractions will never have a single pole. One can then carry out the same manipulation (2.8.3) inside the normal ordering, and so the state with several α oscillators excited comes out as the : : normal-ordered product of the corresponding operators, αµ−m → i ˜ αµ−m → i

2 α 2 α

1/2

1 ∂m X µ (0) , (m − 1)!

m≥1,

(2.8.7a)

1 ∂¯m X µ (0) , (m − 1)!

m≥1.

(2.8.7b)

1/2

Similarly, xµ0 → X µ (0, 0) .

(2.8.8)

Any state can be obtained from |1 by acting with the operators on the left-hand sides of eqs. (2.8.7) and (2.8.8). The operator corresponding to this state is then given by the : : normal-ordered product of the corresponding local operators on the right-hand sides. For example, |0; k ∼ = : eik·X(0,0) : .

(2.8.9)

This is easy to understand: under the translation X µ → X µ + aµ , both the state and the corresponding operator have the same transformation, being multiplied by exp(ik · a). The same method applies to the bc theory. For clarity we specialize to the case λ = 2 which is of interest for bosonic string theory, leaving the generalization to the reader. From the Laurent expansions (2.7.16) and the contour argument it follows that bm |1 = 0 ,

m ≥ −1 ,

cm |1 = 0 ,

m≥2.

(2.8.10)

Note that due to the shifts in the exponents of the Laurent expansion coming from the conformal weights of b and c, the unit operator no longer maps to a ground state. Rather, relations (2.8.10) determine that |1 = b−1 | ↓ .

(2.8.11)

The translation of the raising operators is straightforward, 1 (2.8.12a) ∂m−2 b(0) , m ≥ 2 , (m − 2)! 1 (2.8.12b) c−m → ∂m+1 c(0) , m ≥ −1 . (m + 1)! Notice that the ghost number − 32 state b−1 | ↓ maps to the ghost number 0 unit operator, and the ghost number − 12 state | ↓ to the ghost number 1 operator c. The diﬀerence arises from the nontensor b−m →

66

2 Conformal ﬁeld theory

property (2.5.17) of the ghost number current, (∂z w)jw (w) = jz (z) + q0

q0 ∂z2 w = jz (z) − , ∂z w z

(2.8.13)

where q0 = λ − 12 = 32 . The ghost number of states is conventionally deﬁned by the cylindrical frame expression N g , eq. (2.7.22), while the contour argument of ﬁgure 2.6 relates the ghost number of the vertex operator to the radial frame charge

1 dz jz = N g + q0 . (2.8.14) 2πi This applies to other charges as well; for a tensor current the charges of a state and the corresponding operator are equal. Most of the above can be extended to the βγ theory, but there are certain complications in the superstring so we defer this to chapter 10. All the ideas of this section extend to the open string. The semi-inﬁnite strip Qg ≡

0 ≤ Re w ≤ π ,

Im w ≤ 0 ,

(2.8.15)

maps to a half-disk, the intersection of the upper half-plane and the unit circle, under z = − exp(−iw). The initial state again maps to the origin, which is on the boundary, so there is an isomorphism local operators on the boundary ↔ states on the interval .

(2.8.16)

The details, which are parallel to the above, are left for the reader. Again the doubling trick, extending the ﬁelds into the lower half-plane by holomorphicity, is useful in making the contour arguments. Path integral derivation The state–operator isomorphism is an important but unfamiliar idea and so it is useful to give also a more explicit path integral derivation. Consider a unit disk in the z-plane, with local operator A at the origin and with the path integral ﬁelds φ ﬁxed to some speciﬁc boundary values φb on the unit circle. The path integral over the ﬁelds φi on the interior of the disk with φb held ﬁxed produces some functional ΨA [φb ],

ΨA [φb ] =

[dφi ]φb exp(−S [φi ])A(0) .

(2.8.17)

A functional of the ﬁelds is the Schr¨ odinger representation of a state, so this is a mapping from operators to states. The Schr¨ odinger representation, which assigns a complex amplitude to each ﬁeld conﬁguration, has many uses. It is often omitted from ﬁeld theory texts, the Fock space representation being emphasized.

67

2.8 Vertex operators

To go the other way, start with some state Ψ[φb ]. Consider a path integral over the annular region 1 ≥ |z| ≥ r, with the ﬁelds φb on the outer circle ﬁxed and the ﬁelds φb integrated over the inner circle as follows:

˜

[dφb ][dφi ]φb ,φb exp(−S [φi ]) r−L0 −L0 Ψ[φb ] .

(2.8.18) ˜

That is, the integral is weighted by the state (functional) r−L0 −L0 Ψ[φb ]. Now, the path integral over the annulus just corresponds to propagating from |z| = r to |z| = 1, which is equivalent to acting with the operator ˜ r+L0 +L0 . This undoes the operator acting on Ψ, so the path integral (2.8.18) is again Ψ[φb ]. Now take the limit as r → 0. The annulus becomes a disk, and the limit of the path integral over the inner circle can be thought of as deﬁning some local operator at the origin. By construction the path integral on the disk with this operator reproduces Ψ[φb ] on the boundary. Let us work this out explicitly for a single free scalar ﬁeld X. On the unit circle expand Xb (θ) =

∞

Xn∗ = X−n .

Xn einθ ,

n=−∞

(2.8.19)

The boundary state Ψ[Xb ] can thus be regarded as a function of all of the Xn . Let us ﬁrst identify the state corresponding to the unit operator, given by the path integral without insertion,

Ψ1 [Xb ] =

[dXi ]Xb

1 exp − 2πα

¯ d z ∂X ∂X

2

(2.8.20)

.

Evaluate this by the usual Gaussian method. Separate Xi into a classical part and a ﬂuctuation, Xi = Xcl + Xi , Xcl (z, ¯z ) = X0 +

∞

(2.8.21a)

(z n Xn + ¯z n X−n ) .

(2.8.21b)

n=1

With this deﬁnition, Xcl satisﬁes the equation of motion and Xi vanishes on the boundary. The path integral then separates,

Ψ1 [Xb ] = exp(−Scl ) with

[dXi ]Xb =0

1 exp − 2πα

2

¯

d z ∂X ∂X

,

(2.8.22)

∞ 1 Scl = mnXm X−n d2 z z m−1 ¯z n−1 2πα m,n=1 |z|<1

=

∞ 1 mXm X−m . α m=1

(2.8.23)

68

2 Conformal ﬁeld theory

The Xi integral is a constant, independent of the boundary condition, so

∞ 1 Ψ1 [Xb ] ∝ exp − mXm X−m α m=1

.

(2.8.24)

This is a Gaussian, and is in fact the ground state. To see this, write the raising and lowering operators in the Schr¨ odinger basis,

α in αn = − 1/2 X−n − i 2 (2α )

1/2

∂ , ∂Xn

(2.8.25a)

α 1/2 ∂ in ˜αn = − 1/2 Xn − i , (2.8.25b) 2 ∂X−n (2α ) as follows from the Laurent expansion at |z| = 1 and the mode algebra. Acting on (2.8.24), we ﬁnd αn Ψ1 [Xb ] = ˜αn Ψ1 [Xb ] = 0 ,

n≥0,

(2.8.26)

so this is indeed the ground state. Thus, |1 ∝ |0; 0 .

(2.8.27)

Rather than try to keep track of the overall normalization of the path integral at this point, we deﬁne |1 = |0; 0 . Another easy calculation is the state corresponding to ∂k X; this just adds a factor of ∂k Xcl (0) = k!Xk to the result, so |∂k X = k!Xk Ψ1 = −i

1/2 α

2

(k − 1)!α−k |0; 0 ,

(2.8.28)

and similarly for ∂¯k X. This extends to all products of exponentials and derivatives of X. The conformal normal ordering just cancels the eﬀect of the Xi path integral. 2.9

More on states and operators The OPE

In this section we make additional applications of the state–operator correspondence. The ﬁrst is a simple and general derivation of the OPE, as shown in ﬁgure 2.7. Consider the product Ai (z, ¯z )Aj (0, 0) ,

(2.9.1)

where |z| < 1. We can divide the path integral into an integral over ﬁelds φi on the interior of the unit disk, an integral over ﬁelds φb on the unit circle, and an integral over ﬁelds φe on the exterior of the unit disk. This sort of cutting open of path integrals is discussed in detail in the

2.9 More on states and operators

69

A i(z,z)

ψij,z,z

A j(0,0)

(b)

(a)

A ij,z,z

(c) Fig. 2.7. (a) World-sheet with two local operators. (b) Integration over ﬁelds on the interior of the disk produces boundary state |ψij,z,¯z . (c) Sewing in a disk with the corresponding local operator. Expanding in operators of deﬁnite weight gives the OPE.

appendix. As in the discussion at the end of the previous section, the integral over φi leaves some functional of φb ; call it Ψij,z,¯z [φb ]. By the state–operator correspondence this is equivalent to gluing back the disk with the appropriate local operator Aij,z,¯z (0, 0) and so we are done! To ˜ 0 eigenstates, put this in standard form expand in a complete set of L0 , L Aij,z,¯z =

˜

˜

˜

z hk −hi −hj ¯z hk −hi −hj ckij Ak ,

(2.9.2)

k

with the z- and ¯z -dependence determined by the conformal weights as in eq. (2.4.20). The convergence of the OPE is just the usual convergence of a complete set in quantum mechanics. The construction is possible as long as there are no other operators with |z | ≤ |z|, so that we can cut on a circle of radius |z| + .

70

2 Conformal ﬁeld theory m

i i

Σl

m

=

l j

k

Σl

l

k

j

Fig. 2.8. Schematic picture of OPE associativity.

For three operators Ai (0, 0)Aj (1, 1)Ak (z, ¯z ) ,

(2.9.3)

the regions of convergence of the z → 0 and z → 1 OPEs, |z| < 1 and |1 − z| < 1, overlap. The coeﬃcient of Am in the triple product can then be written as a sum involving clik cmlj or as a sum involving cljk cmli . Associativity requires these sums to be equal. This is represented schematically in ﬁgure 2.8. The Virasoro algebra and highest weight states Consider now the argument of ﬁgure 2.6 for the Virasoro generators:

dz m+1 Lm |A ∼ z T (z)A(0, 0) = 2πi ∼ = Lm · A(0, 0) .

(2.9.4)

Here we have introduced a new idea and notation. Given the state– operator isomorphism, every operator acting on the Hilbert space has an image acting on the space of local operators. In other words, form the corresponding contour around the operator and evaluate the resulting local operator using the OPE. Corresponding to the Lm that map states to states are the Lm · that map local operators to local operators. In general there will be local operators at various positions zi , and there will be a diﬀerent copy of the Virasoro algebra for each one, obtained by a Laurent expansion centered on the position of the operator. In some geometries there may also be a standard set of generators, such as the ones centered on z = 0 for the cylinder. The ‘·’ serves as a reminder that we are talking about the Laurent coeﬃcients around that particular operator. In this notation the T A OPE is T (z)A(0, 0) =

∞ m=−∞

z −m−2 Lm · A(0, 0) .

(2.9.5)

2.9 More on states and operators

71

To relate to the earlier notation (2.4.11), A(n) = Ln−1 · A for n ≥ 0, and the conformal transformation of A is δA(z, ¯z ) = −

∞ 1 n ˜ n−1 · A(z, ¯z ) . ∂ v(z)Ln−1 + (∂n v(z))∗ L n=0

n!

(2.9.6)

Using the general form (2.4.14) for the T A OPE, we have the very useful result ¯ , ˜ −1 · A = ∂A L−1 · A = ∂A , L ˜ 0 · A = ˜hA . L0 · A = hA , L

(2.9.7a) (2.9.7b)

The OPE (2.9.5) implies that a primary ﬁeld O of weight (h, ˜h) corresponds to a state satisfying ˜ 0 |O = ˜h|O , L0 |O = h|O , L ˜ m |O = 0 , m > 0 . Lm |O = L

(2.9.8a) (2.9.8b)

Such a state is conventionally known as a highest weight state. For the ˜ 0 are bounded below. By acting repeatedly with CFTs of interest L0 and L lowering operators on any state we eventually reach a state annihilated by further lowering operators, a highest weight state. An interesting special case is the unit operator. The operator product T 1 is nonsingular, so it follows in any CFT that ˜ m |1 = 0 , Lm |1 = L

m ≥ −1 .

(2.9.9)

As noted in section 2.6, the operators L0 and L±1 form a closed algebra, ˜ ±1 . The full algebra is ˜ 0 and L as do L S L(2, C) .

(2.9.10)

Thus |1 is also called the S L(2, C)-invariant state. It is the only such state because the relations (2.9.7) imply that the operator A corresponding to any S L(2, C)-invariant state is independent of position. It must then be a c-number, as explained after eq. (2.4.24). Unitary CFTs Highest weight states play a special role in string theory, and also in the theory of representations of the Virasoro algebra. We will return to this at various points, but for now we derive a few important general results that hold in unitary CFTs. A unitary CFT is one that has a positive inner product | such that L†m = L−m ,

˜ †m = L ˜ −m . L

(2.9.11)

Recall that the inner product deﬁnes the adjoint via α|Aβ = A† α|β .

72

2 Conformal ﬁeld theory

For example, the X µ CFT is unitary for spacelike µ if we take the inner product

0; k|0; k = 2πδ(k − k )

(2.9.12)

for the ground state and deﬁne α†m = α−m ,

˜α†m = ˜α−m .

(2.9.13)

This implicitly deﬁnes the inner products of all higher states. This CFT is not unitary for µ = 0 because of the opposite sign in the commutator there. The ﬁrst constraint is that any state in a unitary CFT must have h, ˜h ≥ 0. If this is true for highest weight states it is true for all states. For a highest weight state the Virasoro algebra gives 2hO O|O = 2 O|L0 |O = O|[L1 , L−1 ]|O = L−1 |O 2 ≥ 0 ,

(2.9.14)

so hO ≥ 0. It also follows that if hO = ˜hO = 0 then ˜ −1 · O = 0 , L−1 · O = L

(2.9.15)

and so O is independent of position. As noted before, O must then be a c-number. That is, the unit operator is the only (0,0) operator in a unitary CFT. Curiously, X µ itself is the one notable exception: the corresponding state xµ |0; 0 is nonnormalizable because of the inﬁnite range of X µ , and equation (2.9.14) no longer holds. The general theorems are of the most use for the CFTs corresponding to compactiﬁed dimensions, where this kind of exception cannot occur. In a similar way, one ﬁnds that an operator in a unitary CFT is holomorphic if and only if ˜h = 0, and antiholomorphic if and only if h = 0; this is an important result so we repeat it, ∂A = 0 ⇔ h = 0 ,

¯ = 0 ⇔ ˜h = 0 . ∂A

(2.9.16)

Finally, using the above argument with the commutator [Ln , L−n ], one can show that c, ˜c ≥ 0 in a unitary CFT. In fact, the only unitary CFT with c = 0 is the trivial one, Ln = 0. Zero-point energies The state–operator mapping gives a simple alternative derivation of the various normal ordering constants. In any CFT, we know that L0 |1 = 0, and this determines the additive normalization of L0 . In the X CFT, |1 is the ground state |0; 0 , so aX vanishes. In the bc theory, |1 is the excited state b−1 | ↓ , so the weight of | ↓ is −1 in agreement with the earlier result (2.7.21) for λ = 2.

2.9 More on states and operators

73

This also provides one physical interpretation of the central charge. In ˜ 0 = 0. The conformal a unitary CFT the ground state is |1 with L0 = L transformation (2.6.10) between the radial and time-translation generators then implies that E=−

c + ˜c 24

(2.9.17)

for the ground state. This is a Casimir energy, coming from the ﬁnite size of the system, and it depends only on the central charge. On dimensional grounds this energy is inverse to the size of the system, which is 2π here, so the general result would be E=−

π(c + ˜c) . 12

(2.9.18)

We have now given three honest ways of calculating normal ordering constants: from the Virasoro algebra as in eq. (2.7.8), by relating the two forms of normal ordering as in eq. (2.7.11), and from the state–operator mapping above. Nevertheless the idea of adding up zero-point energies is intuitive and a useful mnemonic, so we give a prescription that can be checked by one of the more honest methods: 1. Add the zero-point energies, 12 ω for each bosonic mode and − 12 ω for each fermionic (anticommuting) mode. 2. One encounters divergent sums of the form ∞ n=1 (n−θ), the θ arising when one considers nontrivial periodicity conditions. Deﬁne this to be ∞ n=1

(n − θ) =

1 1 − (2θ − 1)2 . 24 8

(2.9.19)

This is the value one obtains as in eq. (1.3.32) by regulating and discarding the quadratically divergent part. 3. The above gives the normal ordering constant for the w-frame generator T0 , eq. (2.6.8). For L0 we must add the nontensor correction 1 24 c. For the free boson, the modes are integer-valued so we get one-half of 1 . This is just oﬀset the sum (2.9.19) for θ = 0 after step 2, which is − 24 X by the correction in step 3, giving a = 0. For the ghosts we similarly get 2 26 24 − 24 = −1.

74

2 Conformal ﬁeld theory Exercises

2.1 Verify that 1 1 = ∂¯ = 2πδ 2 (z, ¯z ) ¯z z (a) by use of the divergence theorem (2.1.9); (b) by regulating the singularity and then taking a limit. ∂∂¯ ln |z|2 = ∂

2.2 Work out explicitly the expression (2.2.5) for the normal-ordered product of four X µ ﬁelds. Show that it is a harmonic function of each of the positions. 2.3 The expectation value of a product of exponential operators on the plane is n

iki ·X(zi ,¯zi )

:e

:

= iC X (2π)D δ D (

i=1

n i=1 ki )

n

|zij |α ki ·kj ,

i,j=1 i<j

with C X a constant. This can be obtained as a limit of the expectation value (6.2.17) on the sphere, which we will obtain by several methods in chapter 6. (a) Show that the leading behavior as one vertex operator approaches another is in agreement with the OPE (2.2.14). (b) As implied by eq. (2.2.13), the expectation value is |z12 |α k1 ·k2 times a function that is smooth as z1 → z2 . For n = 3, work out the explicit expansion of this smooth function in powers of z12 and ¯z12 . From the ratio of the large-order terms ﬁnd the radius of convergence. (c) Give a general proof that the free-ﬁeld OPE (2.2.4) is convergent inside the dashed circle in ﬁgure 2.1. 2.4 Extend the OPE (2.2.14) up to and including subleading terms of order z 2 , z¯z , and ¯z 2 . Show that the z, ¯z -dependence of each term is in agreement with the general form (2.4.20). Give also the symmetric form as in eq. (2.2.15) and discuss the properties of the coeﬃcients under interchange of the two operators. 2.5 Derive the classical Noether theorem in the form usually found in textbooks. That is, assume that

S [φ] =

dd σ L(φ(σ), ∂a φ(σ))

and ignore the variation of the measure. Invariance of the action implies that the variation of the Lagrangian density is a total derivative, δL = ∂a Ka under a symmetry transformation (2.3.1). Then the classical result, with

75

Exercises conventional string normalization, is

∂L −1 j = 2πi δφα − Ka ∂φα,a a

.

Derive the Lagrangian equations of motion and show that they imply that this current is conserved. Show that it agrees with eq. (2.3.4). 2.6 Consider the ﬂat Euclidean metric δab , in d dimensions. An inﬁnitesimal coordinate transformation δσ a = v a (σ) changes the metric by δgab = −∂a vb − ∂b va . Determine the most general v a (σ) that leaves the metric invariant up to a local rescaling. There are 12 (d + 1)(d + 2) independent solutions in d > 2, and an inﬁnite number in d = 2. [Although the result is given in many places in the literature, the author has not found a simple derivation.] 2.7 (a) By computing the relevant OPEs, conﬁrm the weights stated in eq. (2.4.17) and determine which operators are tensors. (b) Do this for the same operators in the linear dilaton theory. ¯ ν eik·X : ? What are the conditions on 2.8 What is the weight of fµν : ∂X µ ∂X fµν and kµ in order for it to be a tensor? 2.9 Derive the central charges for the linear dilaton, bc, and βγ CFTs by working out the T T OPEs. 2.10 Consider now a patch of world-sheet with boundary. For convenience suppose that the patch lies in the upper half-z-plane, with the real axis being the boundary. Show that expectation values of a normal-ordered ¯ µ (¯z ) :, although ﬁnite in the interior, diverge as operator, say : ∂X µ (z)∂X z approaches the boundary (represent the eﬀect of the boundary by an image charge). Deﬁne boundary normal ordering , which is the same as : : except that the contraction includes the image charge piece as well. Operators on the boundary are ﬁnite if they are boundary normal ordered. In a general CFT on a manifold with boundary, the interior operators and the boundary operators are independent. Label the bases Ai and Br respectively, and deﬁne the former by : : and the latter by . Each set has its own closed OPE: AA → A and BB → B. In addition, the sets are related: Ai , as it approaches the boundary, can be expanded in terms of the Br . Find the leading behaviors of

eik1 ·X(y1 ) eik2 ·X(y2 ) , ik·X(z,¯z )

:e

:,

y1 → y2 (y real) ,

Im(z) → 0 .

The identity (2.7.14) is useful for the latter. 2.11 Evaluate the central charge in the Virasoro algebra for X µ by calcu-

76

2 Conformal ﬁeld theory

lating Lm (L−m |0; 0 ) − L−m (Lm |0; 0 ) . 2.12 Use the OPE and the contour results (2.6.14) and (2.6.15) to derive the commutators (2.7.5) and (2.7.17). 2.13 (a) Show that : b(z)c(z ) : − ◦◦ b(z)c(z )◦◦ =

(z/z )1−λ − 1 z − z

by the method of eq. (2.7.11). (b) Use this to determine the ordering constant in N g , eq. (2.7.22). You also need the conformal transformation (2.8.14). (c) Show that one obtains the same value for N g by a heuristic treatment of the ordering similar to that in section 2.9. 2.14 (a) Determine the ordering constant in Lg0 , eq. (2.7.21), using part (a) of the previous exercise. (b) Determine it using the heuristic rules in section 2.9. 2.15 Work out the state–operator mapping for the open string X µ CFT. 2.16 Let O be a tensor ﬁeld of weight (h, ˜h). Evaluate the commutators [Lm , O(z, ¯z )] using the contour argument of ﬁgure 2.4. Show in particular that the commutator of a (1,1) tensor is a total derivative. 2.17 Extend the argument of eq. (2.9.14) to show that c ≥ 0.

3 The Polyakov path integral

We now begin a systematic study of string theory using the Polyakov path integral and conformal ﬁeld theory. After an introduction to the path integral picture, we discuss gauge ﬁxing, the Weyl anomaly, and vertex operators. We then generalize to strings propagating in curved spacetime. 3.1

Sums over world-sheets

The Feynman path integral is one way to represent a quantum theory, and it is a very natural method for describing interactions in string theory. In path integral quantization, amplitudes are given by summing over all possible histories interpolating between the initial and ﬁnal states. Each history is weighted by exp(iScl /¯h) ,

(3.1.1)

with Scl the classical action for the given history. Thus, one would deﬁne an amplitude in string theory by summing over all world-sheets connecting given initial and ﬁnal curves, as in ﬁgure 3.1(a) for the open string and ﬁgure 3.1(b) for the closed string. The analogous sum for the relativistic point particle produces the free propagator (exercise 5.1). One can imagine a number of ways in which strings might interact. One would be a contact interaction, an energy when two strings intersect. Another would be a long-range force mediated by some quantum ﬁeld. However, it is not possible to add such interactions to string theory in a way that is consistent with the symmetries; we will get some idea later of why this is so. Rather, the only interactions that are allowed are those that are already implicit in the sum over world-sheets. Consider, for example, the world-sheets shown in ﬁgure 3.2. Figure 3.2(a) looks like a quantum correction to the open string amplitude of ﬁgure 3.1(a), involving intermediate states with two open strings. Figure 3.2(b) has three external 77

78

3 The Polyakov path integral

(a)

(b)

Fig. 3.1. (a) An open string world-sheet with the topology of a strip. The heavier curves are the world-lines of string endpoints. (b) A closed string world-sheet with the topology of a cylinder.

(a)

(b)

Fig. 3.2. (a) Quantum correction to open string propagation. (b) Decay of one closed string into two. The dashed lines are time-slices.

closed strings and represents one string decaying into two. It turns out that this is the correct way to introduce interactions into string theory. We will see that these interactions produce a consistent theory, ﬁnite and unitary, which includes gravity. It is interesting to consider the process of ﬁgure 3.2(b) as seen at successive times as shown in ﬁgure 3.3. A closed string splits into two, or in the reverse process two join into one. This is the basic closed string interaction. In closed string theory, all particles are obtained as various states of excitation of the string, and all interactions (gauge, gravitational,

3.1 Sums over world-sheets

79

Fig. 3.3. Successive time-slices from ﬁgure 3.2(b). The arrows indicate the orientation of the string.

(a)

(b)

(c)

(d) Fig. 3.4. Processes involving open strings: (a) open ↔ open + open; (b) closed ↔ open; (c) open + open ↔ open + open; (d) open ↔ open + closed.

and Yukawa, for example) arise from the single process of ﬁgure 3.3. For theories with open strings there are several additional processes that can occur, as shown in ﬁgure 3.4. With a given slicing, the interactions of ﬁgures 3.3 and 3.4 appear to occur at a deﬁnite spacetime point, but this is an illusion. A diﬀerent slicing, in a boosted Lorentz frame, will put the apparent interaction at a diﬀerent point; there is no distinguished point in spacetime. The interaction arises only from the global topology of the world-sheet, while the local properties of the world-sheet are the same as they were in the free case. As discussed in the introduction, it is this smearing out of the interaction that cuts oﬀ the short-distance divergences of gravity.

80

3 The Polyakov path integral

(a)

(b)

Fig. 3.5. Oriented (a) and unoriented (b) contributions to the two-open-string amplitude.

There are actually several diﬀerent string theories, depending on which topologies we include in the sum over world-sheets. First note that the world-sheets we have drawn have two kinds of boundary, the source boundaries corresponding to the initial and ﬁnal string conﬁgurations, and the endpoint boundaries corresponding to the world-lines of the endpoints of open strings. Ignore for now the source boundaries; we will discuss the details of the sources later in this chapter. So in the following discussion ‘boundary’ means endpoint boundary. There are then four ways to deﬁne the sum over world-sheets, which correspond to the four free string theories listed at the end of section 1.4: 1. Closed oriented: All oriented world-sheets without boundary. 2. Closed unoriented: All world-sheets without boundary. 3. Closed plus open oriented: All oriented world-sheets with any number of boundaries. 4. Closed plus open unoriented: All world-sheets with any number of boundaries. There are two things that we should expand upon here. The ﬁrst is the connection between the inclusion of unoriented world-sheets and the Ω = +1 projection described in section 1.4. Figure 3.5 shows two worldsheets that are included in the unoriented theory with open strings. The unoriented world-sheet of ﬁgure 3.5(b) is equivalent to cutting the oriented surface of ﬁgure 3.5(a) along the dashed line, which we will parameterize

3.1 Sums over world-sheets

(a)

81

(b)

Fig. 3.6. (a) An annulus: the intermediate state is two open strings. (b) The same topology drawn as a cylinder: the intermediate state is one closed string.

by 0 ≤ σ ≤ π, and requiring the ﬁelds on the cut edges to satisfy µ µ Xupper (σ) = Xlower (π − σ) .

(3.1.2)

This in turn is equivalent to acting with the operator Ω on the open string state. Adding these two surfaces with appropriate weight then amounts to inserting the operator 12 (1 + Ω), which projects onto states of Ω = +1. Summing over all oriented and unoriented surfaces inserts projection operators on all intermediate states, reducing the spectrum to the Ω = +1 sector. Second, the list above does not include any theory with open strings but not closed. Let us explain in more detail why this is the case. Consider a world-sheet with the topology of an annulus, ﬁgure 3.6(a). For convenience we have drawn only the vacuum amplitude, but by attaching appropriate sources the same argument will also apply to scattering amplitudes. We see from the dashed line that this is a process with an intermediate state of two open strings, and so world-sheets of this topology must be present in any theory with open strings. In ﬁgure 3.6(b) we have drawn the same topology as a cylinder and cut it open on an intermediate state of a single closed string. Thus, even if we start with open strings only, the sum over world-sheets will necessarily include processes in which scattering of open strings produces closed strings. As we develop the amplitudes we will see how this works in detail. The need for closed strings in open string theory can also be seen in another way. Consider the interactions in ﬁgures 3.4(a) and (b), but time reversed: respectively two open strings joining into one, and an open to closed string transition. Near the interaction point these are the same, two endpoints coalescing. To have the ﬁrst interaction without the second would require some nonlocal constraint on the dynamics of the string; this would surely be inconsistent. One can make the same argument with ﬁgures 3.4(c) and (d), where the interaction is locally the reconnection of

82

3 The Polyakov path integral

a pair of strings. So if any open string interaction is allowed, then so is some process in which closed strings are produced from open strings. 3.2

The Polyakov path integral

We now begin to develop the sum over world-sheets, integrating over the ﬁelds of the Polyakov formalism. We make one change from chapter 1: the Minkowskian world-sheet metric γab is replaced with a Euclidean worldsheet metric gab (σ 1 , σ 2 ), of signature (+, +). The integral runs over all Euclidean metrics and over all embeddings X µ (σ 1 , σ 2 ) of the world-sheet in Minkowski spacetime:

[dX dg] exp(−S ) .

(3.2.1)

S = SX + λχ ,

(3.2.2)

The Euclidean action is

with

1 d2 σ g 1/2 g ab ∂a X µ ∂b Xµ , (3.2.3a) SX = 4πα M 1 1 χ= d2 σ g 1/2 R + ds k . (3.2.3b) 4π M 2π ∂M The geodesic curvature k was deﬁned in exercise 1.3; note that the boundary now is always spacelike. The advantage of the Euclidean path integral is that the integral over metrics is better deﬁned. The topologically nontrivial world-sheets we have been describing can have nonsingular Euclidean metrics but not Minkowskian ones. We will take the Euclidean theory (3.2.1) as our starting point, and we will show how to give it a precise deﬁnition and that it deﬁnes a consistent spacetime theory with the advertised properties. However, let us give a brief formal argument that it is equivalent to the Minkowski theory with which we began. We start again with the example of the point particle, where the path integral

[dη dX] exp

i 2

˙ µX ˙ µ − ηm2 dτ η −1 X

(3.2.4)

is oscillatory. The path integral over η and X µ is a product of ordinary integrals, and so we can deform contours just as in ordinary integration. If we take η(τ) → e−iθ η(τ) ,

X 0 (τ) → e−iθ X 0 (τ) ,

(3.2.5)

83

3.2 The Polyakov path integral

for inﬁnitesimal θ, all terms in the exponent acquire a negative real part and so this acts as a convergence factor. Now we can perform a contour rotation in ﬁeld space, all the way to η = −ie, X 0 = −iX D . The integral becomes

1 [de dX] exp − 2

−1

dτ e

D

˙ µX ˙ µ + em2 X

.

(3.2.6)

µ=1

This is the Euclidean point-particle analog of the path integral (3.2.1). We have just made a contour rotation, so the Euclidean path integral gives the same amplitude as the Minkowski one. The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, γab = −e0a e0b + e1a e1b , and make the same rotation on e0a . This provides a formal justiﬁcation for the equivalence of the Minkowski and Euclidean path integrals.1 It has been shown by explicit calculation that they deﬁne the same amplitudes, respectively in the light-cone and conformal gauges. We have in the action (3.2.3a) left the rotation of X 0 implicit, as in the previous chapter, so as to emphasize the Minkowski nature of spacetime. Written in terms of X 0 , the equations of motion and the OPE are covariant, with metric η µν . We have noted that χ is locally a total derivative in two dimensions and therefore depends only on the topology of the world-sheet — it is the Euler number of the world-sheet. Thus, the e−λχ factor in the path integral aﬀects only the relative weighting of diﬀerent topologies in the sum over world-sheets. In particular, if one adds an extra strip to a world-sheet, as we have done in going from ﬁgure 3.1(a) to ﬁgure 3.2(a), the Euler number decreases by one and the path integral is weighted by an extra factor of eλ . Since adding a strip corresponds to emitting and reabsorbing a virtual open string, the amplitude for emitting an open string from any process is proportional to eλ/2 . Adding a handle to any world-sheet reduces the Euler number by 2 and adds a factor e2λ . Since this corresponds to emitting and reabsorbing a closed string, the amplitude for emitting a closed string is proportional to eλ . The Euler term in the action thus controls the coupling constants in the string theory, with go2 ∼ gc ∼ eλ .

(3.2.7)

By the way, λ might seem to be a free parameter in the theory, contradicting the statement in the introduction that there are no such parameters 1

In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.

84

3 The Polyakov path integral

in string theory. This is an important point, and we will resolve it in section 3.7. As an aside, the counting of couplings extends in a simple way to world-sheets with stringy sources. The sources for closed strings are closed boundary loops, while the open string sources are boundary segments. For convenience we will require the open string source boundaries to meet the endpoint boundaries at right angles in the world-sheet metric. We have to be careful because the boundary curvature k diverges at the corners: the Euler number is 1 (3.2.8) χ = ˜χ + nc , 4 where ˜χ includes only the integral over the smooth segments of boundary and nc is the number of corners. The correct weight for the path integral is exp(−λ˜χ) = exp(−λχ + λnc /4) .

(3.2.9)

This follows from unitarity: the λ-dependence must be unchanged if we cut through a world-sheet, leaving fewer internal lines and more external sources. This will be the case for the weight (3.2.9) because the surface integral in ˜ χ cancels on the cut edges. Adding a closed string source amounts to cutting a hole in the world-sheet and decreases χ by 1. Adding an open string source leaves χ unchanged and increases nc by 2. Thus we reach the same conclusion (3.2.7) about the amplitude for emission of closed or open strings. 3.3

Gauge ﬁxing

The path integral (3.2.1) is not quite right. It contains an enormous overcounting, because conﬁgurations (X, g) and (X , g ) that are related to one another by the local diﬀ×Weyl symmetry represent the same physical conﬁguration. We need, in eﬀect, to divide by the volume of this local symmetry group,

[dX dg] exp(−S ) ≡ Z . Vdiﬀ×Weyl

(3.3.1)

We will carry this out by gauge-ﬁxing, integrating over a slice that cuts through each gauge equivalence class once and obtaining the correct measure on the slice by the Faddeev–Popov method. In chapter 1 we imposed light-cone gauge. This is useful for some purposes but hides some of the symmetry of the theory, so we now make a diﬀerent choice. Note that the metric, being symmetric, has three independent components, and that there are three gauge functions, the

3.3 Gauge ﬁxing

85

two coordinates and the local scale of the metric. Thus there is just enough gauge freedom to eliminate the integration over the metric, ﬁxing it at some speciﬁc functional form which we will call the ﬁducial metric, gab (σ) → gˆ ab (σ) .

(3.3.2)

A simple choice is the ﬂat or unit gauge metric gˆ ab (σ) = δab .

(3.3.3)

One sometimes wishes to consider the eﬀect of the diﬀ group alone. In this case, one can bring an arbitrary metric to within a Weyl transformation of the unit form. This is the conformal gauge, gˆ ab (σ) = exp[2ω(σ)]δab .

(3.3.4)

Let us see explicitly that this counting of ﬁelds and gauge symmetries works — that any metric can be brought to the ﬂat form at least locally, in a given neighborhood on the world-sheet. First, make a Weyl transformation to set to zero the Ricci scalar built from gab . The Weyl transformation of the Ricci scalar is g 1/2 R = g 1/2 (R − 2∇2 ω) .

(3.3.5)

To set R = 0 requires solving 2∇2 ω = R as an equation for ω. This is always possible, at least locally, by a standard argument from electrostatics. Now, in two dimensions the vanishing of the Ricci scalar implies the vanishing of the entire Riemann tensor, because the symmetries of the Riemann tensor imply that 1 (3.3.6) Rabcd = (gac gbd − gad gbc )R . 2 So the metric is ﬂat and coordinate-equivalent to the unit metric (3.3.3). Actually, locally there is a little bit of extra gauge freedom. There are diﬀ×Weyl transformations that leave the metric in unit gauge and so are not ﬁxed by the choice of metric. It is convenient here to introduce a complex coordinate z = σ 1 + iσ 2 as in chapter 2, where the ﬂat metric is ds2 = dzd¯z . Consider a coordinate transformation such that z is a holomorphic function of z, z ≡ σ 1 + iσ 2 = f(z) ,

(3.3.7)

combined with a Weyl transformation. The new metric is ds2 = exp(2ω)|∂z f|−2 dz d¯z .

(3.3.8)

ω = ln |∂z f|

(3.3.9)

Then for

86

3 The Polyakov path integral

the metric is invariant. This extra gauge freedom does not conﬂict with our earlier counting (3 = 3) because the mismatch, the set of holomorphic reparameterizations, is of measure zero compared to all reparameterizations. When we consider the full world-sheet, most or all of this extra freedom will be eliminated by the boundary conditions. There are two issues here. First, in the discussion of Noether’s theorem and the Ward identities, we were careful to emphasize below eq. (2.3.10) that these results depend only on the symmetry being deﬁned locally on the world-sheet. The transformation (3.3.7), (3.3.9) will therefore give rise to a conserved current and to Ward identities. This is precisely the conformal invariance studied in chapter 2. We see that it arises as the subgroup of diﬀ×Weyl that leaves invariant the unit metric. The second issue is what happens to our counting of metric and gauge degrees of freedom when we do consider the full world-sheet. We will take this up in chapter 5, where we will see that globally there is a small mismatch between the two, a ﬁnite number of parameters. Until then, we will be concerned only with local properties of the world-sheet theory. The Faddeev–Popov determinant After ﬁxing the metric, the functional integral runs along a slice parameterized by X µ alone. In order to obtain the correct measure, we follow the Faddeev–Popov procedure. The steps are the same as were used to obtain the gauge-ﬁxed measure in Yang-Mills theory, but we will give a self-contained discussion. The idea, illustrated in ﬁgure 3.7, is to separate the path integral into an integral over the gauge group times an integral along the gauge slice, and to divide out the former. The Faddeev–Popov determinant is the Jacobian for this change of variables. Let ζ be shorthand for a combined coordinate and Weyl transformation, ∂σ c ∂σ d gcd (σ) . (3.3.10) ∂σ a ∂σ b Following a standard route, we deﬁne the Faddeev–Popov measure ∆FP by ζ : g → gζ ,

ζ gab (σ ) = exp[2ω(σ)]

1 = ∆FP (g)

[dζ] δ(g − gˆ ζ ) ,

(3.3.11)

where again gˆ ab is the ﬁducial metric. In (3.3.11), [dζ] is a gauge-invariant measure on the diﬀ×Weyl group. We will not need the explicit form of this measure; we will discuss the question of its existence in the next section. ζ (σ) The delta function is actually a delta functional, requiring gab (σ) = gˆ ab at every point.

87

3.3 Gauge ﬁxing

field space gauge slice

gauge orbits Fig. 3.7. Schematic picture of the Faddeev–Popov procedure. The gauge orbits are families of gauge-equivalent conﬁgurations. Integrating over the whole ﬁeld space and dividing by the orbit volume is equivalent to integrating over a slice that intersects each orbit once, with appropriate Jacobian.

Inserting (3.3.11) into the functional integral (3.3.1), the latter becomes

ˆ = Z[g]

[dζ dX dg] ∆FP (g)δ(g − gˆ ζ ) exp(−S [X, g]) . Vdiﬀ×Weyl

(3.3.12)

For later reference, we denote explicitly the dependence of Z on the choice of ﬁducial metric. Carry out the integration over gab , and also rename the dummy variable X → X ζ , to obtain

ˆ = Z[g]

[dζ dX ζ ] ∆FP (gˆ ζ ) exp(−S [X ζ , gˆ ζ ]) . Vdiﬀ×Weyl

(3.3.13)

Now use the gauge invariance2 of ∆FP , of the functional integral measure [dX], and of the action to obtain

ˆ = Z[g]

[dζ dX] ˆ exp(−S [X, g]) ˆ . ∆FP (g) Vdiﬀ×Weyl

(3.3.14)

Finally, nothing in the integrand depends on ζ, so the integral over ζ just produces the volume of the gauge group and cancels the denominator, 2

To show this invariance:

∆FP (g ζ )−1 =

=

[dζ ] δ(g ζ − gˆ ζ ) =

[dζ ] δ(g − gˆ ζ

−1 ·ζ

)

[dζ ] δ(g − gˆ ζ ) = ∆FP (g)−1 ,

where ζ = ζ −1 · ζ . In the second equality we have used the gauge invariance of the delta function, and in the third the invariance of the measure.

88

3 The Polyakov path integral

leaving

ˆ = Z[g]

ˆ exp(−S [X, g]) ˆ . [dX] ∆FP (g)

(3.3.15)

ˆ is the correct measure on the slice. Thus, ∆FP (g) To evaluate the expression (3.3.11) for the Faddeev–Popov measure, let us pretend that the gauge choice precisely ﬁxes the gauge symmetry, so that for exactly one value of ζ the delta function δ(g − gˆ ζ ) is nonzero; for δ(gˆ − gˆ ζ ) this would be the identity. As noted above, there is a small global mismatch, but we will deal with this when the time comes — it does not aﬀect the local properties of the world-sheet. For ζ near the identity, we can expand δgab = 2δω gab − ∇a δσb − ∇b δσa = (2δω − ∇c δσ c )gab − 2(P1 δσ)ab .

(3.3.16)

We have deﬁned a diﬀerential operator P1 that takes vectors into traceless symmetric 2-tensors, 1 (P1 δσ)ab = (∇a δσb + ∇b δσa − gab ∇c δσ c ) . 2

(3.3.17)

Near the identity, the inverse determinant (3.3.11) becomes ˆ −1 ∆FP (g)

=

=

!

[dδω dβ dδσ] exp 2πi

=

ˆ [dδω dδσ] δ −(2δω − ∇·δσ) gˆ + 2Pˆ 1 δσ !

[dβ dδσ] exp 4πi

2

d σ gˆ

1/2 ab

β

ˆ −(2δω − ∇·δσ) gˆ + 2Pˆ 1 δσ

d2 σ gˆ 1/2 β ab (Pˆ 1 δσ)ab

" ab

"

.

(3.3.18)

A hat on a diﬀerential operator indicates that it contains the ﬁducial ˆ In the second equality we have used the integral representation metric g. for the functional delta function, introducing the symmetric tensor ﬁeld β ab . In the third equality we have integrated over δω, which produces a delta functional forcing β ab to be traceless; the functional integral [dβ ] is then over traceless symmetric tensors. We now have a representation ˆ −1 as a functional integral over a vector ﬁeld δσ a and over a for ∆FP (g) traceless symmetric tensor β ab . As discussed in the appendix, we can invert this path integral by replacing each bosonic ﬁeld with a corresponding Grassmann ghost ﬁeld. δσ a → ca , βab → bab ,

(3.3.19a) (3.3.19b)

3.3 Gauge ﬁxing

89

with bab , like β ab , being traceless. Thus,

ˆ = ∆FP (g)

[db dc] exp(−Sg ) ,

(3.3.20)

where the ghost action Sg , with a convenient normalization for the ﬁelds, is 1 ˆ a cb = 1 d2 σ gˆ 1/2 bab ∇ d2 σ gˆ 1/2 bab (Pˆ 1 c)ab . (3.3.21) Sg = 2π 2π Locally on the world-sheet, the Polyakov path integral is now

ˆ = Z[g]

[dX db dc] exp(−SX − Sg ) .

(3.3.22)

The action is quadratic in the ﬁelds, so we can evaluate the path integral in terms of determinants as discussed in the appendix. This step will be carried out in more detail in the chapter 5; the result here is ˆ 2 )−D/2 det Pˆ 1 , ˆ = (det ∇ Z[g] (3.3.23) the ﬁrst determinant coming from the X integration and the second from the ghost integration. In conformal gauge, gˆ ab (σ) = exp[2ω(σ)]δab , the ghost action (3.3.21) is 1 Sg = d2 z bzz ∇¯z cz + b¯z¯z ∇z c¯z 2π 1 = d2 z bzz ∂¯z cz + b¯z¯z ∂z c¯z . (3.3.24) 2π Notice that ω(σ) does not appear in the ﬁnal form. This is because the covariant ¯z -derivative of a tensor that has only z indices reduces to the simple derivative, and vice versa, as the reader can check by working out the connection tensor. We have positioned the indices, lower on b and upper on c, to take advantage of this. Since the action (3.3.24) is Weyl-invariant, it follows that bab and ca must be neutral under the Weyl transformation (in contrast to bab and ca , for example, which contain additional factors of the metric). Since the conformal transformation is a combination of a coordinate transformation (3.3.7) and a Weyl transformation (3.3.9), we can immediately read oﬀ the conformal transformations of bab and ca from the tensor indices: this is a bc CFT with (hb , hc ) = (2, −1) and a ˜b˜c hc ) = (2, −1). CFT with (˜hb , ˜ The derivation that we have given of the gauge-ﬁxed path integral is sometimes referred to as the ‘heuristic’ derivation. The problem is that the starting point, the Polyakov functional integral (3.3.1), is ill deﬁned due to the enormous gauge overcounting. The gauge-ﬁxed functional integral (3.3.22), on the other hand, is quite well deﬁned, and can be calculated explicitly. For practical purposes we should perhaps regard the gaugeﬁxed form as our true starting point, with the gauge-invariant one giving

90

3 The Polyakov path integral

an intuitive interpretation. In the gauge-ﬁxed form, the essential content of the gauge symmetry is still present as BRST invariance, which we will take up in the next chapter. In the open string we need also to consider a patch containing a segment of the boundary. It is convenient to regard functional integrals as being over ﬁelds on a ﬁxed coordinate region, so the diﬀ invariance is limited to coordinate changes that take the boundary into itself. The variation δσ a therefore has vanishing normal component, na δσ a = 0 .

(3.3.25)

This boundary condition is inherited by the corresponding ghost ﬁeld, na ca = 0 ,

(3.3.26)

so ca is proportional to the tangent vector ta . The equations of motion then provide a boundary condition on bab . They have a surface term

∂M

ds na bab δcb = 0 .

(3.3.27)

With the boundary condition on ca , this implies na tb bab = 0 .

(3.3.28)

These are the boundary conditions that we used in section 2.7. 3.4

The Weyl anomaly

A key feature of string theory is that it is not consistent in all spacetime backgrounds, but only in those satisfying certain conditions. For the bosonic string theory in ﬂat spacetime, the condition is D = 26. This was ﬁrst discovered as a pathology of the scattering amplitudes. In the light-cone analysis, it arises as a loss of Lorentz invariance, as we brieﬂy and rather heuristically explained in section 1.3. The underlying source of the restriction is an anomaly in the local world-sheet symmetries. In our present formalism, the anomaly is in the Weyl symmetry: T aa vanishes classically, but not in the quantum theory. The reader who has studied quantum ﬁeld theory has already encountered a related anomaly, although it was likely not described in the same language. A global Weyl transformation, ω(σ) = constant, is equivalent to an overall rescaling of lengths. A number of familiar ﬁeld theories in four dimensions are classically invariant under such a rescaling. These include massless scalar ﬁeld theory with a φ4 interaction, and non-Abelian gauge theory. The scale invariance is evident because the Lagrangian contains no dimensionful parameters — the coupling constant in each case is dimensionless. However, we know that due to divergences in the quantum

3.4 The Weyl anomaly

91

theory there is a nonzero renormalization group beta function in each theory, implying that the eﬀective coupling constant is in fact a function of the length scale. Correspondingly, the trace of the energy-momentum tensor vanishes in the classical theory but is nonzero and proportional to the beta function when quantum eﬀects are taken into account. In the previous section, we ignored the possibility of anomalies. We need to check this: is the gauge-ﬁxed path integral Z[g] really independent of the choice of ﬁducial metric, Z[g ζ ] = Z[g] ?

(3.4.1)

In section 4.2 we will extend this to more general changes of gauge. For convenience, we omit the hat on g henceforth. We will also be interested in path integrals with additional insertions,

. . . g ≡

[dX db dc] exp(−S [X, b, c, g]) . . . .

(3.4.2)

Then we also require

. . . g ζ = . . . g .

(3.4.3)

We are not at the present time interested in the details of the insertions themselves — that will be the subject of the next two sections — so we will restrict attention to gauge transformations ζ vanishing near the positions of the ﬁelds in ‘. . .’. That is, we are asking that Weyl invariance hold as an operator equation. It is easy to preserve the diﬀ and Poincar´e invariances in the quantum theory. For example, one may deﬁne the gauge-ﬁxed path integral using a Pauli–Villars regulator as is done for the harmonic oscillator in section A.1, dividing by the path integral for a massive regulator ﬁeld. The massive regulator ﬁeld can be coupled to the metric in a diﬀ- and Poincar´e-invariant way. However, the diﬀ- and Poincar´e-invariant mass term µ2 d2 σ g 1/2 Y µ Yµ for a regulator ﬁeld Y µ is not Weyl-invariant. We must therefore check Weyl invariance by hand: does the invariance (3.4.3) hold for Weyl transformations? Any Weyl transformation can be obtained as the result of repeated inﬁnitesimal transformations, so it is suﬃcient to study the latter. The energy-momentum tensor T ab is the inﬁnitesimal variation of the path integral with respect to the metric, 1 δ . . . g = − d2 σ g(σ)1/2 δgab (σ) T ab (σ) . . . . (3.4.4) g 4π Classically, T ab comes entirely from the variation of the action, and this coincides with the Noether deﬁnition 4π δ classical S . (3.4.5) T ab (σ) = 1/2 g(σ) δgab (σ)

92

3 The Polyakov path integral

It also reduces to the earlier deﬁnition in the limit of a ﬂat world-sheet. If one takes δgab to have the form of a coordinate transformation, the diﬀ invariance of the path integral implies the conservation of T ab . In the following analysis, we will at ﬁrst neglect possible boundary terms. For a Weyl transformation, the deﬁnition (3.4.4) of T ab means that

1 d2 σ g(σ)1/2 δω(σ) T aa (σ) . . . g , (3.4.6) 2π so Weyl invariance with the general insertions ‘. . .’ can be phrased as the operator statement that the energy-momentum tensor is traceless: δW . . . g = −

?

T aa = 0 .

(3.4.7)

The classical action is Weyl-invariant, but in the quantum theory the trace might be nonzero because we have not managed to ﬁnd a fully gauge-invariant regulator. The trace must be diﬀ- and Poincar´e-invariant, because we have preserved these symmetries, and it must vanish in the ﬂat case because we know from the previous chapter that that theory is conformally invariant. This leaves only one possibility, T aa = a1 R

(3.4.8)

for some constant a1 , with R again the world-sheet Ricci scalar. Terms with more than two derivatives are forbidden for dimensional reasons. In units of world-sheet length, gab and X µ are both dimensionless, so the constant a1 in eq. (3.4.8) is dimensionless as well. Terms with more derivatives would have a coeﬃcient with positive powers of the cutoﬀ length used to deﬁne the path integral, and so vanish in the limit where the cutoﬀ is taken to zero. Calculation of the Weyl anomaly The possible obstruction to a gauge-invariant theory has been reduced to the single number a1 . In fact, this number is proportional to something we have already calculated, the central charge c of the CFT on a ﬂat world-sheet. Both a1 and c are related to the two-point function of the energy-momentum tensor, though to diﬀerent components. To get the precise relation between them we will need to use the diﬀ invariance, as follows. In conformal gauge, a1 (3.4.9) Tz¯z = gz¯z R . 2 Taking the covariant derivative, a1 a1 ∇¯z T¯z z = ∇¯z (gz¯z R) = ∂z R , (3.4.10) 2 2

93

3.4 The Weyl anomaly

where we have used the fact that the metric is covariantly constant. Then by conservation of Tab , a1 (3.4.11) ∇z Tzz = −∇¯z T¯z z = − ∂z R . 2 To ﬁx a1 we compare the Weyl transformations on the left and right. The Weyl transformation (3.3.5) of the right is a1 ∂z ∇2 δω ≈ 4a1 ∂z2 ∂¯z δω ,

(3.4.12)

where we have expanded around a ﬂat world-sheet. To get the Weyl transformation of Tzz we use ﬁrst the conformal transformation (2.4.24), c (3.4.13) −1 δTzz (z) = − ∂z3 v z (z) − 2∂z v z (z)Tzz (z) − v z (z)∂z Tzz (z) . 12 From the discussion in section 3.3, this conformal transformation consists of a coordinate transformation δz = v plus a Weyl transformation 2δω = ∂v + (∂v)∗ . The last two terms in the variation are the coordinate transformation of the tensor, so the Weyl transformation, to leading order around ﬂat space, is c (3.4.14) δW Tzz = − ∂z2 δω . 6 Acting with ∂z = 2∂¯z and comparing with the transformation (3.4.12) gives c c = −12a1 , T aa = − R . (3.4.15) 12 Let us derive this again by a slightly longer route, along which we obtain a useful intermediate result. In conformal gauge, R = −2 exp(−2ω)∂a ∂a ω , ∇2 = exp(−2ω)∂a ∂a .

(3.4.16a) (3.4.16b)

By the contraction with two lowered indices we mean δ ab ∂a ∂b . The Weyl variation (3.4.6) of Z[g] becomes

a1 (3.4.17) Z[exp(2ω)δ] d2 σ δω∂a ∂a ω , π where ‘exp(2ω)δ’ denotes conformal gauge. This integrates immediately to δW Z[exp(2ω)δ] =

Z[exp(2ω)δ] = Z[δ] exp −

a1 2π

d2 σ ∂a ω∂a ω

.

(3.4.18)

Since every metric is diﬀ-equivalent to a conformal metric, eq. (3.4.18) actually determines the complete metric-dependence of Z[g]. We need to ﬁnd the diﬀ-invariant expression that reduces to eq. (3.4.18) in conformal gauge. Using the relations (3.4.16), one veriﬁes that the desired expression

94

3 The Polyakov path integral

is

a1 Z[g] = Z[δ] exp 8π

2

d σ

2 1/2

d σg

R(σ)G(σ, σ )g

1/2

R(σ ) ,

(3.4.19)

where g(σ)1/2 ∇2 G(σ, σ ) = δ 2 (σ − σ )

(3.4.20)

deﬁnes the scalar Green’s function. This is an interesting result: the path integral is completely determined by the anomaly equation. Now expand Z[g] around a ﬂat background, gab = δab + hab , and keep terms of second order in h¯z¯z . The Ricci scalar is 4∂z2 h¯z¯z to ﬁrst order in h¯z¯z , so (ignoring contact terms, where z = z )

Z[δ + h] a1 ln ≈ 2 d2 z d2 z (∂z2 ln |z − z |2 )h¯z¯z (z, ¯z )∂z2 h¯z¯z (z , ¯z ) Z[δ] 8π 3a1 h¯z¯z (z, ¯z )h¯z¯z (z , ¯z ) = − 2 d2 z d2 z . (3.4.21) 4π (z − z )4 We can also calculate this by using second order perturbation theory in the metric, which by eq. (3.4.4) gives 1 Z[δ + h] ≈ 2 ln Z[δ] 8π

2

d z

d2 z h¯z¯z (z, ¯z )h¯z¯z (z , ¯z ) Tzz (z)Tzz (z )

δ

.

(3.4.22) Now use the standard T T OPE (2.4.25). All terms except the ﬁrst involve operators of nonzero spin and so vanish by rotational invariance, leaving

Tzz (z)Tzz (z ) δ = 12 c(z − z )−4 . Comparison with the result (3.4.21) from the Weyl anomaly gives again eq. (3.4.15). Discussion The world-sheet theory consists of the X µ , with central charge D, and the ghosts, whose central charge (2.5.12) is −26 for λ = 2, giving c = cX + cg = D − 26 .

(3.4.23)

The theory is Weyl-invariant only for D = 26, the same as the condition for Lorentz invariance found in chapter 1. When the Weyl anomaly is nonvanishing, diﬀerent gauge choices are inequivalent, and as with anomalous gauge theories any choice leads to a pathology such as loss of covariance or loss of unitarity. In light-cone gauge, for example, choosing diﬀerent spacetime axes requires a change of gauge, so the underlying Weyl anomaly is translated into a Lorentz anomaly. There is another thing that one might try: ignore the Weyl invariance and treat only the diﬀ invariance as a gauge symmetry. The metric would then have one real degree of freedom ω(σ), to be integrated over rather

3.4 The Weyl anomaly

95

than gauge-ﬁxed. This is particularly plausible because the quantum eﬀects in eq. (3.4.18) have generated an action for ω which looks just like that for one of the X µ ; for D < 26 the sign is that of a spacelike dimension. However, the physics is a little exotic because the diﬀ transformation (and therefore the energy-momentum tensor) of ω is diﬀerent from that of the X µ . In fact the theory is the linear dilaton CFT. So there is no (D + 1)-dimensional Lorentz invariance and this theory is not useful for our immediate purpose, which is to understand physics in our part of the universe. It is an interesting model, however, and we will return to this point brieﬂy in section 9.9. We have found above that a ﬂat world-sheet CFT can be coupled to a curved metric in a Weyl-invariant way if and only if the total central charge c is zero. There is a paradox here, however, because the same argument applies to the antiholomorphic central charge ˜c, and we have seen from the example of the bc CFT that c need not equal ˜c. The point is that we have implicitly assumed world-sheet diﬀ invariance in eqs. (3.4.11) and (3.4.19). If c = ˜c there is no diﬀ-invariant expression that agrees with both the O(h2zz ) and O(h¯2z¯z ) calculations. The CFT then cannot be coupled to a curved metric in a diﬀ-invariant way; there is a diﬀ or gravitational anomaly. The paradox shows that c = ˜c is necessary for a CFT to be free of gravitational anomalies, and it can also be shown that this is suﬃcient. Inclusion of boundaries Let us ﬁrst return to eq. (3.4.8) and consider the more general possibility T aa = a1 R + a2 .

(3.4.24)

This is allowed by diﬀ invariance but would be nonvanishing in the ﬂat limit. Earlier we set a2 to zero by the choice (2.3.15b) of Tab , explained in the footnote below that equation. Equivalently, the existence of more than one conserved Tab on the ﬂat world-sheet means that there is some freedom in how we couple to a curved metric. In particular, we can add to the action

Sct = b

d2 σ g 1/2 ,

(3.4.25)

which is a two-dimensional cosmological constant. This is not Weylinvariant. It adds 2πbgab to Tab and so the trace (3.4.24) becomes T aa = a1 R + (a2 + 4πb) .

(3.4.26)

We can therefore make the second term vanish by setting b = −a2 /4π; this is what we have done implicitly in eq. (2.3.15b). The separate values of a2 and b depend on deﬁnitions and are uninteresting.

96

3 The Polyakov path integral

No such local counterterm can remove a1 . For example, d2 σ g 1/2 R is Weyl-invariant, as we have already noted. So if a1 is nonzero, there is a real anomaly in the Weyl symmetry. We now consider the eﬀects of boundaries. Including boundary terms, the most general possible variation is3 1 δW ln . . . g = − d2 σ g 1/2 (a1 R + a2 )δω 2π M 1 − ds (a3 + a4 k + a5 na ∂a )δω , (3.4.27) 2π ∂M where again k = −ta nb ∇a tb is the geodesic curvature of the boundary. The possible counterterms are

Sct =

M

d2 σ g 1/2 b1 +

∂M

ds (b2 + b3 k) .

(3.4.28)

The b3 term is in addition to the geodesic curvature term in the Euler number, in eq. (3.2.3b). The Weyl variation of Sct , being careful to include the dependence of ds and of the unit vectors t and n on the metric, is

δW Sct = 2

M

d2 σ g 1/2 b1 δω +

∂M

ds (b2 + b3 na ∂a )δω .

(3.4.29)

We see that b1 , b2 , and b3 can be chosen to set a2 , a3 , and a5 to zero, leaving a1 and a4 as potential anomalies. There is one more tool at our disposal, the Wess–Zumino consistency condition. Take the second functional variation δW1 (δW2 ln . . . g ) a1 a4 2 1/2 2 = d σ g δω2 ∇ δω1 − ds δω2 na ∂a δω1 π M 2π ∂M a1 2a1 − a4 =− d2 σ g 1/2 ∂a δω2 ∂a δω1 + ds na δω2 ∂a δω1 . π M 2π ∂M (3.4.30) This is a second (functional) derivative and so by deﬁnition must be symmetric in δω1 and δω2 . The ﬁrst term is symmetric but the second one is not, and so a4 = 2a1 . It follows that boundaries, with open string boundary conditions, give rise to no new Weyl anomalies. Weyl invariance holds in the quantum theory if and only if a1 = −c/12 vanishes. This is determined by the local physics in the interior of the world-sheet. The Wess–Zumino consistency condition is a very powerful restriction on the form of possible anomalies. A second application enables us to show that the central charge is a constant. This followed above from 3

We require all terms to be independent of the ﬁelds X µ , as can be shown to hold in free ﬁeld theory. When we consider the interacting case in section 3.7, the Weyl anomaly will take a more general form.

3.5 Scattering amplitudes

97

Lorentz invariance and dimensional analysis, but in more general CFTs one might imagine that C(σ) R(σ) , (3.4.31) 12 where C(σ) is some local operator. In particular, the trace would still vanish on a ﬂat world-sheet. Repeating the calculation (3.4.30) with −C(σ)/12 in place of a1 , there is an additional asymmetric term proportional to ∂a C(σ) from the integration by parts. The consistency condition thus implies that expectation values involving C(σ) are position-independent, implying that C(σ) itself is just a numerical constant c. T aa (σ) = −

3.5

Scattering amplitudes

The idea of a sum over all world-sheets bounded by given initial and ﬁnal curves seems like a natural one. However, it is diﬃcult to deﬁne this in a way that is consistent with the local world-sheet symmetries, and the resulting amplitudes are rather complicated. There is one special case where the amplitudes simplify. This is the limit where the string sources are taken to inﬁnity. This corresponds to a scattering amplitude, an S-matrix element, with the incoming and outgoing strings speciﬁed. For most of this book we will conﬁne ourselves to the S-matrix and similar ‘on-shell’ questions. This is not completely satisfying, because one would like to be able to discuss the state of the system at ﬁnite times, not just asymptotic processes. For now we focus on the S-matrix because it is well understood, while it is not clear what is the correct way to think about string theory oﬀ-shell. We will return to this issue brieﬂy in the next section, and again in chapter 9. So we are now considering a process like that shown in ﬁgure 3.8(a), where the sources are pulled oﬀ to inﬁnity. We will give a heuristic argument for how these sources should be represented in the path integral. Away from the scattering process the strings propagate freely. Each of the incoming and outgoing legs is a long cylinder, which can be described with a complex coordinate w, − 2πt ≤ Im w ≤ 0 ,

w∼ = w + 2π .

(3.5.1)

The lower end of the cylinder, Im w = −2πt, is the end with the external source, the upper end ﬁts onto the rest of the world-sheet, and the circumference is the periodic Re w direction. The limit corresponding to the scattering process is t → ∞. It may seem that we are confusing a long distance in spacetime with a long cylinder in the world-sheet coordinate, but this will turn out to be correct: we will see later that propagation

98

3 The Polyakov path integral

(a)

(b)

Fig. 3.8. (a) Scattering of four closed strings, with sources approaching X 0 = ±∞. (b) Conformally equivalent picture of the four-closed-string scattering: a sphere with small holes.

over long spacetime distances comes precisely from world-sheets where a cylinder is growing long in the sense above. As we know from chapter 2, the cylinder has a conformally equivalent description in terms of the coordinate z, z = exp(−iw) ,

exp(−2πt) ≤ |z| ≤ 1 .

(3.5.2)

In this picture, the long cylinder is mapped into the unit disk, where the external string state is a tiny circle in the center. In the geometry of the internal metric, the process of ﬁgure 3.8(a) now looks like ﬁgure 3.8(b). In the limit t → ∞, the tiny circles shrink to points and the world-sheet reduces to a sphere with a point-like insertion for each external state. The same idea holds for external open string states. The long strips in the process of ﬁgure 3.9(a) can be described as the region − 2πt ≤ Im w ≤ 0 ,

0 ≤ Re w ≤ π ,

(3.5.3)

where Im w = −2πt is the source and Re w = 0, π are endpoint boundaries. Under z = − exp(−iw), this maps into the intersection of the unit disk with the upper half-plane, with a tiny semi-circle cut out at the origin, exp(−2πt) ≤ |z| ≤ 1 ,

Im z ≥ 0 .

(3.5.4)

99

3.5 Scattering amplitudes

(a)

(b)

Fig. 3.9. (a) Scattering of four open strings, with sources approaching X 0 = ±∞. (b) Conformally equivalent picture: a disk with small dents.

The scattering process now looks like ﬁgure 3.9(b). In the limit t → ∞, the sources shrink to points located on the (endpoint) boundary. This is the state–operator mapping, which we have already seen. Each string source becomes a local disturbance on the world-sheet. To a given incoming or outgoing string, with D-momentum k µ and internal state j, there corresponds a local vertex operator Vj (k) determined by the limiting process. Incoming and outgoing states are distinguished by the sign of k 0 ; for an incoming state k µ = (E, k) and for an outgoing state k µ = −(E, k). Figures 3.8 and 3.9 depict only the lowest order amplitudes, but the construction is clearly general: we may restrict attention to compact world-sheets, with no tubes going oﬀ to inﬁnity but with a pointlike insertion in the interior for each external closed string and a pointlike insertion on the boundary for each external open string. An n-particle S-matrix element is then given by Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

n [dX dg] exp(−SX − λχ) d2 σi g(σi )1/2 Vji (ki , σi ) . Vdiﬀ×Weyl i=1

(3.5.5)

To make the vertex operator insertions diﬀ-invariant, we have integrated

100

(a)

3 The Polyakov path integral

(b)

(c)

Fig. 3.10. Compact connected closed oriented surfaces of genus (a) 0, (b) 1, (c) 2.

them over the world-sheet. In the remainder of volume one, we will see that our guess is correct and that this does indeed deﬁne a sensible string S-matrix. Depending on which of the four types of string theories we are considering, the sum over topologies may include unoriented world-sheets, and/or world-sheets with boundary. In general, the sum over topologies is not restricted to connected world-sheets; the total process may involve two or more widely separated sets of particles scattering independently. It is convenient to focus on the connected S-matrix, given by restricting the sum to connected surfaces. To obtain the connected S-matrix, then, we must sum over all compact, connected topologies for the world-sheet. In two dimensions, the classiﬁcation of these topologies is well known. Any compact, connected, oriented two-dimensional surface without boundary is topologically equivalent to a sphere with g handles; g is known as the genus of the surface. In ﬁgure 3.10 we show g = 0, 1, and 2. Boundaries can be added by cutting holes in a closed surface. Any compact, connected, oriented two-dimensional surface is topologically equivalent to a sphere with g handles and b holes. For example, (g, b) = (0, 1) is a disk, (0, 2) is an annulus, and (0, 3) is a pair of pants. To describe unoriented surfaces, it is useful to introduce the crosscap: cut a hole in the surface and identify diametrically opposite points. In more detail, take a complex coordinate z, cut out a disk of slightly less than unit radius, and glue the opposite points together by deﬁning pairs of points with z = −1/¯z to be equivalent. The resulting surface is unoriented because the gluing is antiholomorphic. Also, unlike the case of a boundary, the cross-cap introduces no edge or other local feature: the edge introduced by the cutting is only the boundary of a coordinate patch. Any compact connected closed surface, oriented or unoriented, can be obtained by adding g handles and c cross-caps to the sphere; any compact connected surface can be obtained by adding g handles, b boundaries, and c cross-caps to the sphere. Actually, these descriptions

3.6 Vertex operators

101

are somewhat redundant: in either the case with boundary or the case without, one obtains every topology exactly once by restricting the sum to let only one of g and c be nonzero. For example, (g, b, c) = (0, 0, 1) is the projective plane, (g, b, c) = (0, 1, 1) is the M¨ obius strip, (g, b, c) = (0, 0, 2) is the Klein bottle, and a torus with cross-cap can be obtained either as (g, b, c) = (0, 0, 3) or as (g, b, c) = (1, 0, 1), trading two cross-caps for a handle. The Euler number is given by χ = 2 − 2g − b − c .

(3.5.6)

The number of distinct topologies is very small compared to the number of distinct Feynman graphs at a given order in ﬁeld theory. For example, in the closed oriented theory, there is exactly one topology at each order of perturbation theory. In a ﬁeld theory, the number of graphs grows factorially with the order. The single string graph contains all the ﬁeld theory graphs. In various limits, where handles become long compared to their circumferences, we can approximate the handles by lines and the string graph in that limit is approximated by the corresponding Feynman graph.

3.6

Vertex operators

Using the state–operator mapping, the vertex operator for the closed string tachyon is

V0 = 2gc → gc

d2 σ g 1/2 eik·X d2 z : eik·X : .

(3.6.1)

We have now included a factor gc in the mapping. This is the closed string coupling constant, from adding an extra string to the process: we use the normalization of the vertex operator as the deﬁnition of the coupling. In the second line we have gone to the ﬂat world-sheet. The vertex operator must be (diﬀ×Weyl)-invariant, so in particular it must be conformally invariant on the ﬂat world-sheet. To oﬀset the transformation of d2 z, the operator must be a tensor of weight (1, 1). By a straightforward OPE calculation, eik·X is a tensor of weight h = ˜h = α k 2 /4, so the condition is m2 = −k 2 = −

4 . α

(3.6.2)

This is precisely the mass found in the light-cone quantization. Similarly, the tensor states at the ﬁrst excited level of the closed string

102

3 The Polyakov path integral

have the ﬂat world-sheet vertex operators

2gc ¯ ν eik·X : . d2 z : ∂X µ ∂X (3.6.3) α The normalization relative to the tachyon vertex operator follows from the state–operator mapping, and we will see in chapter 6 that the same relative normalization is obtained from unitarity of the S-matrix. The weight is α k 2 h = ˜h = 1 + , (3.6.4) 4 so these are massless, again agreeing with the light-cone quantization. There are further conditions in order that this be a tensor, as we will work out below. Vertex operators in the Polyakov approach We have in the state–operator mapping a systematic means of writing the ﬂat world-sheet vertex operator for any state. Since the world-sheet can always be made locally ﬂat, this is in principle all we need. For perspective, however, it is useful to study the curved world-sheet vertex operators in the Polyakov formalism. We will do this in the remainder of the section. The method we use will be somewhat clumsy and not as systematic as the state–operator mapping, but some of the results are useful. The reader need not work out all the details. Any operator to be included in the Polyakov path integral (3.5.5) must respect the local diﬀ×Weyl symmetry of the theory. We have not been speciﬁc in the ﬁrst line of eq. (3.6.1) as to how the operator is to be deﬁned on the curved world-sheet. As in the earlier discussion of the Weyl anomaly, it is convenient to use a method that preserves diﬀ invariance automatically, and to check the Weyl transformation by hand. For most purposes dimensional regularization is used in the literature. However, we will not have extensive need for this and so will not introduce it; rather, we will generalize the normal ordering introduced earlier. Deﬁne a renormalized operator [ F ]r by

1 [ F ]r = exp 2

δ δ d σd σ ∆(σ, σ ) µ F. δX (σ) δXµ (σ ) 2

2

(3.6.5)

Here α (3.6.6) ln d2 (σ, σ ) , 2 where d(σ, σ ) is the geodesic distance between points σ and σ . As with normal ordering, the expression (3.6.5) instructs us to sum over all ways of ∆(σ, σ ) =

103

3.6 Vertex operators

contracting pairs in F using ∆(σ, σ ). On the ﬂat world-sheet, d2 (σ, σ ) = |z − z |2 and this reduces to conformal normal ordering, which we have studied in detail. On a curved world-sheet it cancels the singular part of the self-contractions of ﬁelds in F. Diﬀ invariance is obvious, but the contraction depends on the metric and so introduces the Weyl dependence 1 2

δ δ [ F ]r , µ δX (σ) δXµ (σ ) (3.6.7) the ﬁrst term being the explicit Weyl dependence in the operator. A vertex operator for a state of momentum k µ must transform under translations in the same way as the state, and so will be of the form eik·X times derivatives of X µ . Operators with diﬀerent numbers of derivatives will not mix under Weyl transformations, so we can start by considering zero derivatives, the operator (3.6.1). The Weyl variation comes from the explicit factor of g 1/2 as well as from the renormalization as given in eq. (3.6.7), δW [ F ] r = [ δW F ] r +

δW V0 = 2gc

d2 σd2 σ δW ∆(σ, σ )

d2 σ g 1/2 2δω(σ) −

k2 δW ∆(σ, σ) [ eik·X(σ) ]r . 2

(3.6.8)

At short distance, d2 (σ, σ ) ≈ (σ − σ )2 exp(2ω(σ))

(3.6.9)

and so α ln(σ − σ )2 . 2 The Weyl variation is nonsingular in the limit σ → σ, ∆(σ, σ ) ≈ α ω(σ) +

(3.6.10)

δW ∆(σ, σ) = α δω(σ) . The condition for the Weyl variation (3.6.8) to vanish is then as already deduced above.

(3.6.11) k2

= 4/α ,

Oﬀ-shell amplitudes? Observe that a naive attempt to deﬁne oﬀ-shell amplitudes, by taking k µ oﬀ the mass shell, is inconsistent with the local world-sheet symmetries. In position space this means that a local probe of the string world-sheet, given by inserting δ (X(σ) − x0 ) = D

dD k exp[ik · (X(σ) − x0 )] (2π)D

(3.6.12)

into the path integral, is inconsistent because it involves all momenta. From several points of view this is not surprising. First, the argument given above that we could use pointlike sources (vertex operators) for

104

3 The Polyakov path integral

string states required a limiting process that essentially restricts us to on-shell questions. Second, string theory contains gravity, and in general relativity simple oﬀ-shell amplitudes do not exist. This is because we have to specify the location of the probe, but the coordinates are unphysical and coordinate-invariant oﬀ-shell quantities are much more complicated. Third, in string theory we do not have the freedom to introduce additional ﬁelds to measure local observables (analogous to the way electroweak processes are used to probe strongly interacting systems): we must use the strings themselves, or other objects that we will discuss (D-branes and solitons) that are intrinsic to the theory. One can deﬁne oﬀ-shell amplitudes, at least in perturbation theory, if one ﬁxes a gauge (light-cone gauge being simplest for this) and uses stringy sources. The S-matrix formula (3.5.5) can then be derived in a way more akin to the reduction formula from quantum ﬁeld theory, where it is possible to deﬁne ﬁnite-time transition amplitudes and then take the inﬁnite-time limit. Incidentally, the reader who is trying to relate the present discussion to ﬁeld theory should note that the path integral expression (3.5.5) is not analogous to a Green’s function but will in fact have the properties of an S-matrix element. The analogous object in ﬁeld theory is the Green’s function with its external propagators amputated and its external momenta restricted to the mass shell. The discussion of the local probe (3.6.12) also shows the problem with a contact interaction between strings. The simplest form for such an interaction would be

2

d σ g(σ) M

1/2

M

d2 σ g(σ )1/2 δ D (X(σ) − X(σ )) ,

(3.6.13)

which is nonvanishing whenever the world-sheet self-intersects. However, the delta function involves all momenta, as in eq. (3.6.12), so this is not Weyl-invariant. The structure of string theory is quite rigid: strings can couple to strings only in the way we have described at the beginning of this chapter. Massless closed string vertex operators It is interesting, if a little messy, to carry the Polyakov treatment of vertex operators one level further. There is no way to make a world-sheet scalar with exactly one derivative, so the next case is two derivatives. The diﬀ-invariant possibilities are gc V1 = α

#

d2 σ g 1/2 (g ab sµν + iab aµν )[ ∂a X µ ∂b X ν eik·X ]r $

+ α φR [ eik·X ]r ,

(3.6.14)

3.6 Vertex operators

105

where sµν , aµν , and φ are respectively a symmetric matrix, an antisymmetric matrix, and a constant. The antisymmetric tensor ab is normalized g 1/2 12 = 1, or g 1/2 z¯z = −i. The i accompanying the antisymmetric tensor in the vertex operator can be understood as arising from the Euclidean continuation, because this term necessarily has an odd number of time derivatives (one) . This result applies generally to Euclidean actions and vertex operators. To extend the analysis of Weyl invariance we need to work out the Weyl dependence (3.6.11) of the geodesic distance to higher order. One ﬁnds 1 (3.6.15a) ∂a δW ∆(σ, σ ) = α ∂a δω(σ) , σ =σ 2 1+γ ∂a ∂b δW ∆(σ, σ ) = (3.6.15b) α ∇a ∂b δω(σ) , σ =σ 2 γ ∇a ∂b δW ∆(σ, σ ) = − α ∇a ∂b δω(σ) . (3.6.15c) σ =σ 2 Here, γ = − 23 , but we have left it as a parameter for later reference; the third equation can be obtained from the second and the gradient of the ﬁrst. Using eq. (3.3.5) for the variation of the curvature and eqs. (3.6.7) and (3.6.15) for the variation of the renormalization gives # gc d2 σ g 1/2 δω (g ab Sµν + iab Aµν )[ ∂a X µ ∂b X ν eik·X ]r δW V1 = 2 $ +α FR [ eik·X ]r ,

(3.6.16)

where Sµν = −k 2 sµν + kν k ω sµω + kµ k ω sνω − (1 + γ)kµ kν sωω + 4kµ kν φ , (3.6.17a) 2 ω ω Aµν = −k aµν + kν k aµω − kµ k aνω , (3.6.17b) 1 1 F = (γ − 1)k 2 φ + γk µ k ν sµν − γ(1 + γ)k 2 sνν . (3.6.17c) 2 4 In deriving the Weyl variation (3.6.16) we have integrated by parts and also used the relation α γ [ ∇2 X µ eik·X ]r = i k µ R [ eik·X ]r . (3.6.18) 4 The left-hand side would vanish by the naive equation of motion, but here the equation of motion multiplies another operator at the same point. The general principle that holds in this case is that the operator need not vanish but it is not independent — it can be expanded in terms of the other local operators in the theory. In general the coeﬃcients of these operators depend on the renormalization scheme; the result (3.6.18) can be obtained by taking the Weyl variation of each side.

106

3 The Polyakov path integral

Since Sµν , Aµν , and F multiply independent operators, the condition for Weyl invariance is Sµν = Aµν = F = 0 .

(3.6.19)

In order to get an accurate count of the number of operators, we must note that not all V1 of the form (3.6.14) are independent. Rather, under (3.6.20a) sµν → sµν + ξµ kν + kµ ξν , aµν → aµν + ζµ kν − kµ ζν , (3.6.20b) γ (3.6.20c) φ → φ+ k·ξ , 2 the change in V1 integrates to zero using the equation of motion (3.6.18). Choose now a vector nµ satisfying n · k = 1 and n2 = 0. A complete set of independent operators is obtained by restricting to nµ sµν = nµ aµν = 0 .

(3.6.21)

This is 2D equations for the 2D parameters ξµ and ζµ ; one can show that one equation and one parameter are trivial, but that eqs. (3.6.21) are otherwise nondegenerate and so deﬁne an independent set of vertex operators. By using condition (3.6.21) and solving ﬁrst Sµν nµ nν = 0, then Sµν nµ = Aµν nµ = 0, and ﬁnally the full Sµν = Aµν = F = 0, one ﬁnds that k2 = 0 , (3.6.22a) ν µ k sµν = k aµν = 0 , (3.6.22b) 1+γ µ φ= (3.6.22c) sµ . 4 Again we ﬁnd a mass-shell condition (3.6.22a), this time corresponding to the ﬁrst excited states of the closed string. There is also the condition (3.6.22b) that the polarization be transverse to the momentum, as required for a physical polarization of a massless tensor ﬁeld. In the frame in which 1 k µ = (1, 1, 0, 0, . . . , 0) , nµ = (−1, 1, 0, 0, . . . , 0) , (3.6.23) 2 conditions (3.6.21) and (3.6.22b) imply that all 0- and 1-components vanish, while eq. (3.6.22c) ﬁxes φ. This leaves precisely (D − 2)2 independent operators, the same number of massless closed string states as found in the light-cone gauge. The condition nµ sµν = nµ aµν = 0 is needed to remove zero-norm lightlike polarizations. However, the introduction of the vector nµ spoils manifest Lorentz invariance, and the theory is Lorentz-invariant only because diﬀerent choices of nµ are equivalent due to (3.6.20). The equivalence relations (3.6.20) are therefore essential in order for the theory to be

107

3.6 Vertex operators

Lorentz-invariant with a positive Hilbert space norm. These relations are the signatures of local spacetime symmetries. In particular, ξµ is an inﬁnitesimal spacetime coordinate transformation, and ζµ is the local spacetime symmetry of the antisymmetric tensor ﬁeld. We now see in the interacting theory the existence of local spacetime symmetries, which we argued in section 1.4 had to be present because of the existence of massless spin-1 and spin-2 ﬁelds. There is a technical point we should mention. Diﬀerent renormalization schemes assign diﬀerent names to the renormalized operators. The most common scheme is dimensional regularization; the operators in this scheme are related to those in the scheme above by [ eik·X ]DR = [ eik·X ]r , [ ∂a X µ eik·X ]DR = [ ∂a X µ eik·X ]r ,

(3.6.24a) (3.6.24b)

[ ∂a X µ ∂b X ν eik·X ]DR = [ ∂a X µ ∂b X ν eik·X ]r − [ ∇a ∂b X µ eik·X ]DR = [ ∇a ∂b X µ eik·X ]r + i

α gab η µν R[ eik·X ]r , 12 (3.6.24c)

α gab k µ R[ eik·X ]r . 12

(3.6.24d)

Comparing with eq. (3.6.18), we see that in dimensional regularization γ = 0, so the equations of motion hold inside the regulated operator. This is a convenience — many equations simplify. In particular, in the spacetime gauge transformation (3.6.20), φ is invariant. Open string vertex operators The extension to the open string is straightforward. We leave the details for the reader as exercise 3.9, giving only some results. The tachyon vertex operator is

go

∂M

ds [ eik·X ]r

(3.6.25)

and is Weyl-invariant for k 2 = 1/α . The photon vertex operator is go − i 1/2 eµ (2α )

∂M

˙ µ eik·X ]r , ds [ X

(3.6.26)

where the normalization relative to the tachyon is obtained either from the state–operator mapping or unitarity, the i is as discussed for the closed string vertex operator (3.6.14), and the sign is a matter of convention. This operator is Weyl-invariant if k 2 = 0 and k · e = 0. There is an equivalence eµ ∼ = eµ + λkµ , which is a spacetime gauge transformation. This leaves D − 2 transverse polarizations.

108

3 The Polyakov path integral 3.7

Strings in curved spacetime

Now we are ready to consider an important new issue, strings moving in a curved spacetime. Let us recall ﬁrst that the point-particle action has a natural extension to curved spacetime. Replacing the ﬂat metric ηµν with a general metric Gµν (x) gives the action Spp =

1 2

˙ ν − ηm2 . ˙ µX dτ η −1 Gµν (X)X

(3.7.1)

After eliminating η this becomes the invariant proper time along the particle world-line. Its variation is well known to give the geodesic equation, representing the motion of a particle in a gravitational ﬁeld. Making the same replacement in the Polyakov action gives Sσ =

1 4πα

M

d2 σ g 1/2 g ab Gµν (X)∂a X µ ∂b X ν .

(3.7.2)

This is a natural guess, but the reader might make the following objection. We have learned that the graviton itself is one state of the string. A curved spacetime is, roughly speaking, a coherent background of gravitons, and therefore in string theory it is a coherent state of strings. Merely putting a curved metric in the action (3.7.2) doesn’t seem stringy enough. To see why it is justiﬁed, consider a spacetime that is close to ﬂat, Gµν (X) = ηµν + χµν (X)

(3.7.3)

with χµν small. The integrand in the world-sheet path integral is

1 exp(−Sσ ) = exp(−SP ) 1 − 4πα

2

d σg M

1/2 ab

µ

ν

g χµν (X)∂a X ∂b X + . . .

.

(3.7.4) The term of order χ is precisely the vertex operator for the graviton state of the string, eq. (3.6.14) with χµν (X) = −4πgc eik·X sµν .

(3.7.5)

So indeed, inserting the curved metric does seem to make contact with what we already know. The action (3.7.2) can be thought of as describing a coherent state of gravitons by exponentiating the graviton vertex operator. This suggests a natural generalization: include backgrounds of the other massless string states as well. From the form of the vertex operators, eq. (3.6.14), one has Sσ =

1 4πα

M

d2 σ g 1/2

g ab Gµν (X) + iab Bµν (X) ∂a X µ ∂b X ν + α RΦ(X) . (3.7.6)

3.7 Strings in curved spacetime

109

The ﬁeld Bµν (X) is the antisymmetric tensor, and the dilaton involves both Φ and the diagonal part of Gµν , as implied by the equation of motion (3.6.22c). As a check, let us see that the spacetime gauge invariances are respected. Under a change of variables in the path integral corresponding to a ﬁeld redeﬁnition X µ (X), the action (3.7.6) is invariant with Gµν and Bµν transforming as tensors and Φ as a scalar. From the spacetime point of view, this is a coordinate transformation. The action is also invariant under δBµν (X) = ∂µ ζν (X) − ∂ν ζµ (X) ,

(3.7.7)

which adds a total derivative to the Lagrangian density. This is a generalization of the electromagnetic gauge transformation to a potential with two antisymmetric indices. The three-index ﬁeld strength Hωµν = ∂ω Bµν + ∂µ Bνω + ∂ν Bωµ

(3.7.8)

is invariant. There is a similar generalization to an antisymmetric n-tensor potential, which plays an important role in the superstring and will be developed in volume two. The theory thus depends only on gauge-invariant objects built out of the metric and other ﬁelds. One can think of the ﬁelds X µ as being coordinates on a manifold. This is called the target space, because the X µ deﬁne an embedding, world-sheet → target .

(3.7.9)

In string theory, the target space is spacetime itself. A ﬁeld theory such as (3.7.2), in which the kinetic term is ﬁeld-dependent and so ﬁeld space is eﬀectively a curved manifold, is known for historic reasons as a nonlinear sigma model. Nonlinear sigma models have had many applications in particle physics and quantum ﬁeld theory. For example, the neutral and charged pion ﬁelds can be regarded to good approximation as coordinates on the group manifold S U(2). The nonlinear sigma model action is no longer quadratic in X µ , so the path integral is now an interacting two-dimensional quantum ﬁeld theory. Expand the path integral around the classical solution X µ (σ) = xµ0 for any chosen point xµ0 . With X µ (σ) = xµ0 + Y µ (σ),

Gµν (X)∂a X µ ∂b X ν = Gµν (x0 ) + Gµν,ω (x0 )Y ω

+ 12 Gµν,ωρ (x0 )Y ω Y ρ + . . . ∂a Y µ ∂b Y ν .

(3.7.10)

The ﬁrst term in this expansion is a quadratic kinetic term for the ﬁeld Y µ . The next term is a cubic interaction, and so forth. The coupling constants in the expansion, Gµν,ω (x0 ) and so on, involve derivatives of

110

3 The Polyakov path integral

the metric at x0 . In a target space with characteristic radius of curvature Rc , derivatives of the metric are of order Rc−1 . The eﬀective dimensionless coupling in the theory is therefore α1/2 Rc−1 . If the radius of curvature Rc is much greater than the characteristic length scale of the string, then this coupling is small and perturbation theory in the two-dimensional ﬁeld theory is a useful tool. Notice that in this same regime we can employ another tool as well. Since the characteristic length scale is long compared to the string, we can ignore the internal structure of the string and use low energy eﬀective ﬁeld theory. String theory enters in determining the eﬀective low energy action for the ﬁelds, as we will soon see. We have also implicitly used α1/2 Rc−1 1 when restricting our attention to massless backgrounds: when wavelengths are long compared to the string scale, massive string states are not created. The nonlinear sigma model (3.7.2) is a renormalizable theory: the ﬁelds Y µ have dimension zero and so the interactions all have dimension two. Nevertheless the couplings, which are the coeﬃcients in the expansion of the metric, are inﬁnite in number (unless some symmetry restricts the form of the metric). In fact it is more useful to think of the metric function itself, rather than its expansion coeﬃcients, as deﬁning the couplings of the theory.

Weyl invariance We have emphasized that Weyl invariance is essential to the consistency of string theory. The action (3.7.6) will deﬁne a consistent string theory only if the two-dimensional quantum ﬁeld theory is Weyl-invariant. This action happens to be the most general action that is classically invariant under a rigid Weyl transformation, with δω independent of σ. This is easy to see: under a rigid Weyl transformation, the scaling of a term with n derivatives will be proportional to 2 − n. Of course, it is necessary also to consider local Weyl transformations, under which the Gµν and Bµν terms in the action (3.7.6) are classically invariant while the Φ term is not, and to include the quantum contributions to the Weyl transformation properties. In the limit that Bµν and Φ are small, and Gµν is close to ηµν , we can use the results of the previous section to ﬁnd the Weyl transformation. Write Sσ = SP − V1 + . . ., with SP the ﬂat-space action and V1 the vertex operator (3.6.14). Then Gµν (X) = ηµν − 4πgc sµν eik·X ,

(3.7.11a)

Bµν (X) = −4πgc aµν eik·X ,

(3.7.11b)

Φ(X) = −4πgc φe

ik·X

.

(3.7.11c)

3.7 Strings in curved spacetime

111

We could of course take linear combinations of vertex operators with diﬀerent momenta. To ﬁrst order the Weyl variation is then given in eq. (3.6.16); we take for convenience a renormalization scheme in which γ = 0. Relating this Weyl transformation to T aa as in eq. (3.4.6) gives 1 G ab i B ab 1 βµν g ∂a X µ ∂b X ν − βµν ∂a X µ ∂b X ν − β Φ R , 2α 2α 2 where, to linear order in χµν , Bµν and Φ, T aa = −

(3.7.12)

α 2 ∂ χµν − ∂ν ∂ω χµω − ∂µ ∂ω χων + ∂µ ∂ν χωω + 2α ∂µ ∂ν Φ , 2 (3.7.13a) α ω B βµν ≈ − ∂ Hωµν , (3.7.13b) 2 D − 26 α 2 βΦ ≈ (3.7.13c) − ∂ Φ. 6 2 We have included in β Φ the ﬂat spacetime anomaly found in section 3.4, including the contribution proportional to 26 from the ghost ﬁelds. The symbol β is used for the coeﬃcients in T aa because these are essentially the renormalization group beta functions, governing the dependence of the physics on world-sheet scale. We will discuss this connection further, though it does not come up in detail until chapter 15. The Weyl anomalies (3.7.13) have further contributions from higher orders in the ﬁelds. For example, expanding the path integral to second order in the cubic interaction from the expansion (3.7.10) will give rise to divergences when the two interaction vertices approach one another. ¯ ν times the Their OPE includes a singularity of the form |z|−2 ∂X µ ∂X 2 square of the derivative of Gαβ . Integrating d z produces a logarithmic divergence. Diﬀ invariance requires that the integral be cut oﬀ in terms of the invariant distance, |z| exp(ω) > a0 . This introduces a dependence G on the scale of the metric, ln zmin = −ω + ln a0 . The Weyl variation βµν 2 thus gets a contribution proportional to O(Gαβ,γ ) which combines with the linear second derivative term to form the spacetime Ricci tensor. We will not work this out in detail, but quote the result keeping all terms with up to two spacetime derivatives G βµν ≈−

G = α Rµν + 2α ∇µ ∇ν Φ − βµν

α Hµλω Hν λω + O(α2 ) , 4

(3.7.14a)

α B βµν = − ∇ω Hωµν + α ∇ω ΦHωµν + O(α2 ) , (3.7.14b) 2 D − 26 α 2 α βΦ = − ∇ Φ + α ∇ω Φ∇ω Φ − Hµνλ H µνλ + O(α2 ) . 6 2 24 (3.7.14c) Several terms in (3.7.14) can be recognized from the linear approximation

112

3 The Polyakov path integral

(3.7.13), but now made covariant under a change of spacetime coordinates. The spacetime Ricci tensor is Rµν , distinguished from the world-sheet Ricci tensor Rab . Terms with more derivatives are higher order in the worldsheet α1/2 Rc−1 expansion. Incidentally, the most eﬃcient method to carry the calculation of the Weyl anomaly to higher order is not the one we have described, but rather dimensional regularization. The condition that the world-sheet theory be Weyl-invariant is thus G B = βµν = βΦ = 0 . βµν

(3.7.15)

G = 0 These are sensible-looking equations of motion. The equation βµν resembles Einstein’s equation with source terms from the antisymmetric B = 0 is the antisymmetric tensor ﬁeld and the dilaton. The equation βµν tensor generalization of Maxwell’s equation, determining the divergence of the ﬁeld strength.

Backgrounds One other qualitative feature of the ﬁeld equations is readily understood: the ﬁeld Φ always occurs diﬀerentiated, so there is an invariance under X µ -independent shifts of Φ. This is because such a shift only changes the world-sheet action (3.7.6) by a term proportional to the Euler number, and so does not aﬀect local properties like Weyl invariance. In particular, the background Gµν (X) = ηµν ,

Bµν (X) = 0 ,

Φ(X) = Φ0

(3.7.16)

is exactly Weyl-invariant for any constant Φ0 . This is just the ﬂat spacetime action (3.2.2), with λ = Φ0 .

(3.7.17)

The value of λ determines the coupling strength between strings, but we now see that this does not mean that there are diﬀerent string theories with diﬀerent values of this parameter. Diﬀerent values of λ correspond not to diﬀerent theories, but to diﬀerent backgrounds in a single theory. The other parameter appearing in the Nambu–Goto and Polyakov actions, the Regge slope α , is also not really a free parameter because it is dimensionful. It simply deﬁnes the unit of length, and can be absorbed in the deﬁnition of X µ. Changing the ﬁelds Gµν , Bµν , and Φ in the world-sheet action would seem, from the two-dimensional point of view, to give a new theory. From the point of view of the full string theory, one is merely looking at a diﬀerent background — a diﬀerent state — in the same theory. This is one of the features that makes string theory attractive. In the Standard Model, and in most attempts to unify the Standard Model within

3.7 Strings in curved spacetime

113

quantum ﬁeld theory, there are many constants that are not determined by the theory; any values for these constants give a consistent theory. In string theory there are no such free parameters — the coupling constants depend on the state and are determined in principle by the dynamics. Of course, for now this only moves the diﬃculty elsewhere, because we do not understand the dynamics well enough to know the values of the background ﬁelds. It is striking that Einstein’s equation turns up in what seems to be a rather out-of-the-way place, as the condition for Weyl invariance of a two-dimensional ﬁeld theory. String theory provides the physical connection between these two ideas. It is also notable that, because of the need for Weyl invariance, strings can propagate consistently only in a background that satisﬁes appropriate ﬁeld equations. This parallels our earlier discovery that only on-shell vertex operators make sense. Another observation: the condition D = 26 came from the R term in T aa . In the nonlinear sigma model this is generalized to β Φ = 0. We see in eq. (3.7.14c) that the leading term in β Φ is proportional to D − 26 but that there are corrections involving the gradients of the ﬁelds. Evidently we can have other values of D if the ﬁelds are not constant. This is true, although we cannot really conclude it from eq. (3.7.14c) because this has been derived in the approximation α1/2 Rc−1 1 where the correction terms in β Φ are small. In fact, exact solutions with D = 26 are known. For now we give one simple example, saving others for later. This is Gµν (X) = ηµν ,

Bµν (X) = 0 ,

Φ(X) = Vµ X µ .

(3.7.18)

The beta functions (3.7.14) vanish if Vµ V µ =

26 − D . 6α

(3.7.19)

This result is actually exact, because the ﬁelds (3.7.18) are constant or linear in X µ , so the world-sheet path integral remains Gaussian. Varying gab to determine Tab for this theory, one ﬁnds that it is none other than the linear dilaton CFT (hence the name), whose central charge c = D + 6α Vµ V µ is indeed 26 under the condition (3.7.19). Because Φ must have a large gradient in order to change D, this CFT does not describe our roughly static and homogeneous four-dimensional spacetime, but it may have cosmological applications. This CFT is also important because the cases D = 1 and D = 2 are simple enough for the string theory to be exactly solvable, as we will discuss in section 9.9.

114

3 The Polyakov path integral The spacetime action

The ﬁeld equations (3.7.15) can be derived from the spacetime action 1 S= 2 2κ0

D

d x (−G)

1/2 −2Φ

e

−

2(D − 26) 1 + R − Hµνλ H µνλ 3α 12

+ 4∂µ Φ∂µ Φ + O(α ) .

(3.7.20)

The normalization constant κ0 is not determined by the ﬁeld equations and has no physical signiﬁcance since it can be changed by a redeﬁnition of Φ. One can verify that δS = −

1 2κ20 α

dD x (−G)1/2 e−2Φ δGµν β Gµν + δBµν β Bµν

+ 2δΦ − 12 Gµν δGµν (β Gωω − 4β Φ ) .

(3.7.21)

The action is written in bold as a reminder that this is the eﬀective action governing the low energy spacetime ﬁelds, unlike the world-sheet actions that have appeared everywhere previously in this book. It is often useful to make a ﬁeld redeﬁnition of the form ˜ µν (x) = exp(2ω(x))Gµν (x) , G

(3.7.22)

which is a spacetime Weyl transformation. The Ricci scalar constructed ˜ µν is from G

˜ = exp(−2ω) R − 2(D − 1)∇2 ω − (D − 2)(D − 1)∂µ ω∂µ ω . R

(3.7.23)

For the special case D = 2, this is the Weyl transformation (3.3.5). Let ω = 2(Φ0 − Φ)/(D − 2) and deﬁne ˜ = Φ − Φ0 , Φ

(3.7.24)

which has vanishing expectation value. The action becomes

˜ ˜ ˜ 1/2 − 2(D − 26) e4Φ/(D−2) d X (−G) +R 3α 1 −8Φ/(D−2) 4 ˜ µνλ µ˜ ˜ ˜ ˜ − e Hµνλ H − ∂µ Φ∂ Φ + O(α ) , 12 D−2

1 S= 2 2κ

D

(3.7.25)

where tildes have been inserted as a reminder that indices here are raised ˜ µν , the gravitational Lagrangian density takes ˜ µν . In terms of G with G 2 . The constant κ = κ eΦ0 is the ˜ ˜ 1/2 R/2κ the standard Hilbert form (−G) 0 gravitational coupling, which in four-dimensional gravity has the value κ = (8πGN )1/2 =

(8π)1/2 = (2.43 × 1018 GeV)−1 . MP

(3.7.26)

3.7 Strings in curved spacetime

115

˜ µν Commonly, Gµν is called the sigma model metric or string metric, and G the Einstein metric. Because of the force from dilaton exchange, there is no equivalence principle and so no way to single out a preferred deﬁnition of the metric. Looking ahead, there are higher order eﬀects in string theory that can give the dilaton a mass and so a ﬁnite range — a good thing, because the equivalence principle does of course hold to good accuracy. Note that the dilaton appears in the action (3.7.20) in an overall factor e−2Φ and otherwise is always diﬀerentiated. This is consistent with the fact that adding an overall constant to the dilaton has no eﬀect on the Weyl anomaly. From the spacetime point of view this multiplies the action by a constant, which does not change the equations of motion. It also reﬂects the fact that in quantum ﬁeld theory it is possible to rescale the ﬁelds in such a way that the coupling g appears only as an overall factor of g −2 , which in string theory is e−2Φ . For example, the Yang–Mills action is usually introduced in the form

1 dD x Tr (Fµν F µν ) , S =− 4 Fµν = ∂µ Aν − ∂ν Aµ − ig[Aµ , Aν ]

(3.7.27a) (3.7.27b)

(matrix notation). By deﬁning gAµ = Aµ ,

gFµν = Fµν ,

(3.7.28)

g is removed from the ﬁeld strength, covariant derivative, and gauge transformation and appears only in an overall g −2 in the action. In string theory this holds when the action is written in terms of the ﬁelds appearing on the string world-sheet. If we make a dilaton-dependent ﬁeld redeﬁnition as in going to the form (3.7.25) then it no longer holds. When g appears only in the normalization of the action, every propagator contains a factor of g 2 and every interaction a factor of g −2 . It is not hard to show that an L-loop contribution to the eﬀective action has exactly L − 1 more propagators than vertices and so scales as g 2(L−1) . In string theory this is e−χΦ , as we would expect. The discussion of backgrounds extends to open strings. We have seen that open string vertex operators are integrated along the boundary, so open string ﬁelds appear as boundary terms in the sigma model action. Compactiﬁcation and CFT The four string theories studied thus far (with or without boundaries, oriented or unoriented) have a number of features in common. One good feature is the automatic inclusion of general relativity. There are also several bad features: the need for 26 dimensions, the existence of the tachyon, and the absence of fermions in the spectrum. The elimination

116

3 The Polyakov path integral

of the tachyon and the introduction of fermions will have to wait until volume two. However, the presence of general relativity provides a natural way to account for the extra dimensions. In general relativity the geometry of spacetime is dynamical, not ﬁxed. Flat spacetime is only one of many solutions to the ﬁeld equations. There may well be solutions in which some dimensions are large and ﬂat and others are small and highly curved. The metric would be

gMN =

ηµν 0

0 gmn (xp )

.

(3.7.29)

We have divided the 26 coordinates M, N = 0, . . . , 25 into four ‘spacetime’ coordinates µ, ν = 0, . . . , 3 and 22 ‘internal’ coordinates m, n, p = 4, . . . , 25. The metric (3.7.29) is ﬂat in the four spacetime dimensions and curved in the 22 internal dimensions. The internal dimensions are assumed to be compact. As one example, the background ﬁeld equations (3.7.15) are satisﬁed to order α if the dilaton ﬁeld is constant, the antisymmetric tensor is zero, and the metric for the internal space is Ricci-ﬂat, Rmn = 0. In other words, spacetime is of the form M d × K, where M d is ddimensional Minkowski space and K is some (26−d)-dimensional compact Riemannian space.4 In such a spacetime, the physics on length scales much longer than the size of K is the same as in a d-dimensional Minkowski space. One says that 26 − d dimensions have been compactiﬁed, though cosmologically it seems more appropriate to say that the large dimensions have decompactiﬁed. As we discussed brieﬂy in the introduction, there are good reasons, independent of string theory, to consider the possibility that our spacetime might be of this form. The compact dimensions must be small enough not to have been observed. In fact, we will see in chapter 8, and in more detail in chapter 18, that the size of any extra dimensions must be within a few orders of magnitude of the Planck length, 1.6 × 10−33 cm. We can also think about generalizing this idea further. The basic physical consistency conditions are few in number. We require Lorentz invariance, and we require that quantum mechanical probabilities be positive and conserved — to date, nothing in string theory seems to require that the standard quantum mechanical framework be modiﬁed, though it would be interesting to learn otherwise. Although these conditions are simple there is a tension between them, in gauge theory and gravitation as well as in string theory. In a manifestly covariant gauge there are negative norm timelike excitations that must decouple from physical amplitudes. In the light-cone gauge the inner product is positive but covariance is no longer manifest. 4

We will generally use D to refer to the total dimension of a string theory and d when we may be talking about a subset of the world-sheet ﬁelds.

3.7 Strings in curved spacetime

117

One necessary condition is therefore two-dimensional diﬀ invariance, so that the unphysical oscillations are merely oscillations of the coordinate system. Weyl invariance is a more technical requirement: the extra degree of freedom that appears when Weyl invariance is lost is less intuitive than an oscillation of the coordinate system, but we have argued in section 3.4 that it is incompatible with Lorentz invariance. We will also make an additional technical assumption: that the embedding time X 0 appears in the world-sheet action only in the term −

1 4πα

M

d2 σ g 1/2 g ab ∂a X 0 ∂b X 0 .

(3.7.30)

For example, in the sigma model this means that the ﬁelds are static with G0µ = η0µ and B0µ = 0. This assumption is not much of a restriction. The form (3.7.30) is suﬃcient if one is interested in static states. Timedependent backgrounds are certainly of interest for cosmological and other reasons, but they can be analyzed using low energy eﬀective ﬁeld theory except in the extreme case that the time scale is of the order of the string scale. The reason for making the assumption (3.7.30) is to put the main problem, the wrong-sign nature of X 0 , in a rather explicit form. The requirement that the local invariances be maintained leads to the following proposal for a more general string theory: Replace the 25 spatial X µ ﬁelds with any unitary CFT having c = ˜c = 25. This ensures that the full two-dimensional theory, with X 0 and the ghosts, can be coupled to a curved metric in a (diﬀ×Weyl)-invariant way. The unitary (positive inner product) condition is necessary because there is only enough local symmetry to remove the unphysical X 0 excitation; the world-sheet inner product is relevant here because it becomes the spacetime inner product in the one-particle sector. We will see in volume two that further generalization is possible, enlarging simultaneously the local symmetry group and the set of unphysical excitations. If we are interested in a bosonic string theory whose low energy physics looks four-dimensional, we keep the free ﬁelds X µ for µ = 0, 1, 2, 3, and replace the remaining X µ with a compact unitary CFT having c = ˜c = 22. By compact, we mean that the spectrum is discrete, just as for a quantum ﬁeld theory in a ﬁnite volume. There have been some eﬀorts to construct all CFTs, or all CFTs having various additional properties. We will describe some of these in chapter 15. Incidentally, all of these theories have tachyons, the product of the unit state |1 from the internal CFT with the ground state |0; k from the four-dimensional CFT. How large a generalization have we just made? If the local symmetries are the only essential restriction, we are free to introduce ﬁelds on the world-sheet with various world-sheet and spacetime quantum numbers. This seems to take us far beyond the original theory of 26 X µ ﬁelds. How-

118

3 The Polyakov path integral

ever, in two dimensions there are many equivalences between seemingly diﬀerent quantum ﬁeld theories, so that the situation is not clear. It is generally assumed that all string theories based on diﬀerent CFTs but the same world-sheet gauge symmetries and topology are diﬀerent ground states of the same theory. This causes a semantic problem: we often refer to strings with diﬀerent world-sheet Lagrangians as diﬀerent theories, though they are most likely just diﬀerent vacua in one theory. We will take this even further in chapter 14, seeing that all string theories appear to be vacua of a single theory. As we develop the techniques of string theory in volume one, we will usually take the simplest case of ﬂat 26-dimensional spacetime in explicit examples. However, many results will be developed in such a way as to apply to the very general string theory we have described above. Exercises 3.1 (a) Evaluate the Euler number χ for a ﬂat disk and for a disk with the metric of a hemisphere. (b) Show that the Euler number of a sphere with b holes is 2 − b. You may take any convenient metric on the sphere. (c) Extend this to the result 2 − b − 2g for the sphere with b holes and g handles. 3.2 (a) Show that an n-index traceless symmetric tensor in two dimensions has exactly two independent components. The traceless condition means that the tensor vanishes when any pair of indices is contracted with g ab . (b) Find a diﬀerential operator Pn that takes n-index traceless symmetric tensors into (n + 1)-index traceless symmetric tensors. (c) Find a diﬀerential operator PnT that does the reverse. (d) For u an (n + 1)-index traceless symmetric tensor and v an n-index traceless symmetric tensor, show that with appropriate normalization (u, Pn v) = (PnT u, v). The inner product of traceless symmetric tensors of any rank is (t, t ) =

d2 σ g 1/2 t · t ,

the dot denoting contraction of all indices. 3.3 (a) Work out the form of the covariant derivative in conformal gauge. Show in particular that ∇z reduces to ∂z when acting on a tensor component with all ¯z indices, and vice versa. Since raising or lowering reverses z and ¯z indices, this means that it is always possible to work with simple derivatives by appropriately positioning the indices. (b) Use this to give simple expressions for Pn and PnT . 3.4 To gain familiarity with the Faddeev–Popov procedure, it is useful to

Exercises

119

consider it in a more general and abstract context. This is described in the ﬁrst page of section 4.2 (which is independent of section 4.1). Derive the general result (4.2.3). 3.5 (A continuation of the previous problem.) Consider two diﬀerent gauge choices, F A (φ) = 0 and GA (φ) = 0. The Faddeev–Popov procedure shows that the gauge-ﬁxed path integrals (4.2.3) are both equal to the original gauge-invariant path integral and so equal to each other. However, this passes through the somewhat heuristically deﬁned gauge-invariant path integral. Adapt the Faddeev–Popov procedure to show directly the equivalence of the two gauge-ﬁxed path integrals. This is what is relevant in practice, showing for example that manifestly covariant and manifestly unitary gauges deﬁne the same theory, which therefore has both properties. In chapter 4 we will show this equivalence by a more abstract method, BRST symmetry. 3.6 On dimensional grounds, a current that is conserved on a ﬂat worldsheet has the possible anomaly ∇a j a = aR . By comparing the Weyl transformations of the two sides, show that the ˜ ˜ is 4a. sum of the coeﬃcients of z −3 in T j and of ¯z −3 in T 3.7 In our second calculation of the Weyl anomaly, we have in a roundabout way calculated a Feynman graph with a loop of scalars or ghosts and with two external gravitational ﬁelds. Do the calculation for scalars directly, in momentum space. You will need to use your favorite invariant regulator: dimensional, Pauli–Villars, or zeta function. Show that the result agrees with eq. (3.4.19). The linearized expression for the curvature can be conveniently read from eqs. (3.7.13a) and (3.7.14a). (Warning: the full calculation is a bit lengthy.) 3.8 Verify the Weyl transformation (3.6.16) of the vertex operator V1 . 3.9 Find the Weyl invariance conditions for the open string tachyon and photon vertex operators. It is simplest to take a coordinate system in which the boundary is a straight line. Recall from exercise 2.10 that the propagator includes an image charge term, so it is eﬀectively doubled for ﬁelds directly on the boundary. 3.10 Find the general vertex operator and the conditions for Weyl invariance at the level m2 = 1/α of the open string. Find a complete set of independent Weyl-invariant operators and compare with the states found in the light-cone quantization. [Answer: there appears to be one extra operator. In fact its amplitudes vanish, but we need the formalism of the next chapter to understand the reason, that it corresponds to a null state.] G by the method suggested above 3.11 Calculate the order H 2 term in βµν

120

3 The Polyakov path integral

eq. (3.7.14). You need precisely the term (z¯z )−1 in the OPE; more singular terms require a shift in the tachyon background, but this is an artifact of the bosonic string. G = β B = 0, the nonlinear sigma model will be conformally 3.12 If βµν µν invariant on a ﬂat world-sheet. According to the discussion at the end of section 3.4, in this case β Φ must be independent of X. Show this for the form (3.7.14); you will need the Bianchi identities.

3.13 Consider a spacetime with d ﬂat dimensions and 3 dimensions in the shape of a 3-sphere. Let H be proportional to the completely antisymmetric tensor on the 3-sphere and let the dilaton be constant. Using the form (3.7.14) for the equations of motion, show that there are solutions with d + 3 = 26. These solutions are outside the range of validity of eq. (3.7.14), but we will see in chapter 15 that there are exact solutions of this form, though with H limited to certain quantized values. 3.14 Verify that the background (3.7.18) gives the linear dilaton energymomentum tensor (2.5.1).

4 The string spectrum

4.1

Old covariant quantization

In conformal gauge, the world-sheet ﬁelds are X µ and the Faddeev–Popov ghosts bab and ca . The Hilbert space is bigger than the actual physical spectrum of the string: the D sets of αµ oscillators include unphysical oscillations of the coordinate system, and there are ghost oscillators. As is generally the case in covariant gauges, there are negative norm states from the timelike oscillators (because the commutator is proportional to the spacetime metric ηµν ), and also from the ghosts. The actual physical space is smaller. To see how to identify this smaller space, consider the amplitude on the inﬁnite cylinder for some initial state |i to propagate to some ﬁnal state |f . Suppose that we have initially used the local symmetries to ﬁx the metric to a form gab (σ). Consider now a diﬀerent gauge, with metric gab (σ) + δgab (σ). A physical amplitude should not depend on this choice. Of course, for a change in the metric we know how the path integral changes. From the deﬁnition (3.4.4) of T ab ,

1 d2 σ g(σ)1/2 δgab (σ) f|T ab (σ)|i . (4.1.1) 4π In order that the variation vanish for arbitrary changes in the metric, we need δ f|i = −

ψ|T ab (σ)|ψ = 0

(4.1.2)

for arbitrary physical states |ψ , |ψ . There is another way to think about this. The original equation of motion from variation of gab was T ab = 0. After gauge-ﬁxing, this does not hold as an operator equation: we have a missing equation of motion because we do not vary gab in the gauge-ﬁxed theory. The condition (4.1.2) says that the missing equation of motion must hold for matrix elements between physical states. When we vary the gauge we must take into 121

122

4 The string spectrum

account the change in the Faddeev–Popov determinant, so the energymomentum tensor in the matrix element is the sum of the X µ and ghost contributions, g X + Tab . Tab = Tab

(4.1.3)

The X µ may be replaced by a more general CFT (which we will refer to as the matter CFT), in which case g m + Tab . Tab = Tab

(4.1.4)

In the remainder of this section we will impose the condition (4.1.2) in a simple but somewhat ad hoc way. This is known as the old covariant quantization (OCQ), and it is suﬃcient for many purposes. In the next section we will take a more systematic approach, BRST quantization. These are in fact equivalent, as will be shown in section 4.4. In the ad hoc approach, we will simply ignore the ghosts and try to restrict the matter Hilbert space so that the missing equation of motion m = 0 holds for matrix elements. In terms of the Laurent coeﬃcients Tab ˜m this is Lm n = 0, and in the closed string also Ln = 0. One might ﬁrst try m to require physical states to satisfy Ln |ψ = 0 for all n, but this is too strong; acting on this equation with Lm m and forming the commutator, one encounters an inconsistency due to the central charge in the Virasoro algebra. However, it is suﬃcient that only the Virasoro lowering and zero operator annihilate physical states, (Lm n + Aδn,0 )|ψ = 0

for n ≥ 0 .

(4.1.5)

For n < 0 we then have m

ψ|Lm n |ψ = L−n ψ|ψ = 0 .

(4.1.6)

m Lm† n = L−n

(4.1.7)

We have used

as follows from the Hermiticity of the energy-momentum tensor. The conditions (4.1.5) are consistent with the Virasoro algebra. At n = 0 we have as usual included the possibility of an ordering constant. A state satisfying (4.1.5) is called physical. In the terminology of eq. (2.9.8), a physical state is a highest weight state of weight −A. The condition (4.1.5) is similar to the Gupta–Bleuler quantization of electrodynamics. Using the adjoint (4.1.7), one sees that a state of the form |χ =

∞

Lm −n |χn

(4.1.8)

n=1

is orthogonal to all physical states for any |χn . Such a state is called spurious. A state that is both physical and spurious is called null. If |ψ

123

4.1 Old covariant quantization

is physical and |χ is null, then |ψ + |χ is also physical and its inner product with any physical state is the same as that of |ψ . Therefore these two states are physically indistinguishable, and we identify |ψ ∼ (4.1.9) = |ψ + |χ . The real physical Hilbert space is then the set of equivalence classes, HOCQ =

Hphys . Hnull

(4.1.10)

Let us see how this works for the ﬁrst two levels of the open string in ﬂat spacetime, not necessarily assuming D = 26. The only relevant terms are 2 Lm 0 = α p + α−1 · α1 + . . . ,

Lm ±1

1/2

= (2α )

p · α±1 + . . . .

(4.1.11a) (4.1.11b)

At the lowest mass level, the only state is |0; k . The only nontrivial 2 condition at this level is (Lm 0 + A)|ψ = 0, giving m = A/α . There are no null states at this level — the Virasoro generators in the spurious state (4.1.8) all contain raising operators. Thus there is one equivalence class, corresponding to a scalar particle. At the next level there are D states |e; k = e · α−1 |0; k .

(4.1.12)

e; k|e; k = 0; k|e∗ · α1 e · α−1 |0; k

= 0; k|(e∗ · e + e∗ · α−1 e · α1 )|0; k

= eµ∗ eµ (2π)D δ D (k − k ) .

(4.1.13)

µ αµ† n = α−n ,

(4.1.14)

The norm is

We have used as follows from the Hermiticity of X µ , and also

0; k|0; k = (2π)D δ D (k − k ) ,

(4.1.15)

as follows from momentum conservation. The timelike excitation has a negative norm. The Lm 0 condition gives 1+A . α The other nontrivial physical state condition is m2 =

Lm 1 |e; k ∝ p · α1 e · α−1 |0; k = e · k|0; k = 0

(4.1.16)

(4.1.17)

124

4 The string spectrum

and so k · e = 0. There is a spurious state at this level: 1/2 Lm k · α−1 |0; k . −1 |0; k = (2α )

(4.1.18)

That is, eµ ∝ k µ is spurious. There are now three cases: (i) If A > −1, the mass-squared is positive. Going to the rest frame, the physical state condition removes the negative norm timelike polarization. The spurious state (4.1.18) is not physical, k · k = 0, so there are no null states and the spectrum consists of the D − 1 positive-norm states of a massive vector particle. (ii) If A = −1, the mass-squared is zero. Now k · k = 0 so the spurious state is physical and null. Thus, (4.1.19) k · e = 0 , eµ ∼ = eµ + γkµ . This describes the D − 2 positive-norm states of a massless vector particle. (iii) If A < −1, the mass-squared is negative. The momentum is spacelike, so the physical state condition removes a positive-norm spacelike polarization. The spurious state is not physical, so we are left with a tachyonic vector particle with D − 2 positive-norm states and one negative-norm state. Case (iii) is unacceptable. Case (ii) is the same as the light-cone quantization. Case (i) is not the same as the light-cone spectrum, having diﬀerent masses and an extra state at the ﬁrst level, but does not have any obvious inconsistency. The diﬃculty is that there is no known way to give such a theory consistent interactions. Case (ii) is the one of interest in string theory, and the one we will recover from the more general approach of the next section. The result at the next level is quite interesting: it depends on the constant A and also on the spacetime dimension D. Restricting to the value A = −1 found at the ﬁrst excited level, the spectrum only agrees with the light-cone spectrum if in addition D = 26. If D > 26, there are negative-norm states; if D < 26 the OCQ spectrum has positive norm but more states than in light-cone quantization. The derivation is left to the reader. The OCQ at A = −1 and D = 26 is in fact the same as light-cone quantization at all mass levels, HOCQ = Hlight-cone ,

(4.1.20)

as will be shown in section 4.4. It is only in this case that consistent interactions are known. The extension to the closed string is straightforward. There are two sets of oscillators and two sets of Virasoro generators, so at each level the

4.1 Old covariant quantization

125

spectrum is the product of two copies of the open string spectrum (except for the normalization of p2 in Lm 0 ). The ﬁrst two levels are then 4 ; α eµν αµ−1 ˜ αν−1 |0; k , m2 = 0 , k µ eµν = k ν eµν = 0 , eµν ∼ = eµν + aµ kν + kµ bν , a · k = b · k = 0 . |0; k ,

m2 = −

(4.1.21a) (4.1.21b) (4.1.21c)

The relevant values are again A = −1, D = 26. As in light-cone quantization, there are (D − 2)2 massless states forming a traceless symmetric tensor, an antisymmetric tensor, and a scalar.

Mnemonic For more general string theories, it is useful to have a quick mnemonic for obtaining the zero-point constant. As will be derived in the following sections, the Lm 0 condition can be understood as g (Lm 0 + L0 )|ψ, ↓ = 0 .

(4.1.22)

That is, one includes the ghost contribution with the ghosts in their ground state | ↓ , which has Lg0 = −1. The L0 generators diﬀer from the Hamiltonian by a shift (2.6.10) proportional to the central charge, but the total central charge is zero in string theory so we can just as well write this condition as (H m + H g )|ψ, ↓ = 0 .

(4.1.23)

Now apply the mnemonic for zero-point energies given at the end of chapter 2. In particular, the ghosts always cancel the µ = 0, 1 oscillators because they have the same periodicities but opposite statistics. So the rule is that A is given as the sum of the zero-point energies of the transverse oscillators. This is the same rule as in the light-cone, giving 1 ) = −1. here A = 24(− 24 Incidentally, for the purposes of counting the physical states (but not for their precise form) one can simply ignore the ghost and µ = 0, 1 oscillators and count transverse excitations as in the light-cone. The condition (4.1.22) requires the weight of the matter state to be 1. Since the physical state condition requires the matter state to be a highest weight state, the vertex operator must then be a weight 1 or (1,1) tensor. This agrees with the condition from section 3.6 that the integrated vertex operator be conformally invariant, which gives another understanding of the condition A = −1.

126

4 The string spectrum 4.2

BRST quantization

We now return to a more systematic study of the spectrum. The condition (4.1.2) is not suﬃcient to guarantee gauge invariance. It implies invariance for arbitrary ﬁxed choices of gab , but this is not the most general gauge. In light-cone gauge, for example, we placed some conditions on X µ and some conditions on gab . To consider the most general possible variation of the gauge condition, we must allow δgab to be an operator, that is, to depend on the ﬁelds in the path integral. In order to derive the full invariance condition, it is useful to take a more general and abstract point of view. Consider a path integral with a local symmetry. The path integral ﬁelds are denoted φi , which in the present case would be X µ (σ) and gab (σ). Here we use a very condensed notation, where i labels the ﬁeld and also represents the coordinate σ. The gauge invariance is α δα , where again α includes the coordinate. By assumption the gauge parameters α are real, since we can always separate a complex parameter into its real and imaginary parts. The gauge transformations satisfy an algebra (4.2.1) [δα , δβ ] = f γαβ δγ . Now ﬁx the gauge by conditions F A (φ) = 0 ,

(4.2.2)

where once again A includes the coordinate. Following the same Faddeev– Popov procedure as in section 3.3, the path integral becomes

[dφi ] exp(−S1 ) → Vgauge

[dφi dBA dbA dcα ] exp(−S1 − S2 − S3 ) ,

(4.2.3)

where S1 is the original gauge-invariant action, S2 is the gauge-ﬁxing action S2 = −iBA F A (φ) ,

(4.2.4)

and S3 is the Faddeev–Popov action S3 = bA cα δα F A (φ) .

(4.2.5)

We have introduced the ﬁeld BA to produce an integral representation of the gauge-ﬁxing δ(F A ). There are two things to notice about this action. The ﬁrst is that it is invariant under the Becchi–Rouet–Stora–Tyutin (BRST) transformation δB φi = −icα δα φi , δB BA = 0 , δB bA = BA , i δB cα = f αβγ cβ cγ . 2

(4.2.6a) (4.2.6b) (4.2.6c) (4.2.6d)

4.2 BRST quantization

127

This transformation mixes commuting and anticommuting objects, so that must be taken to be anticommuting. There is a conserved ghost number, which is +1 for cα , −1 for bA and , and 0 for all other ﬁelds. The original action S1 is invariant by itself, because the action of δB on φi is just a gauge transformation with parameter icα . The variation of S2 cancels the variation of bA in S3 , while the variations of δα F A and cα in S3 cancel. The second key property is that δB (bA F A ) = i(S2 + S3 ) .

(4.2.7)

Now consider a small change δF in the gauge-ﬁxing condition. The change in the gauge-ﬁxing and ghost actions gives δ f|i = i f|δB (bA δF A )|i

= − f| { QB , bA δF A } |i ,

(4.2.8)

where we have written the BRST variation as an anticommutator with the corresponding conserved charge QB . Therefore, physical states must satisfy

ψ|{ QB , bA δF A }|ψ = 0 .

(4.2.9)

In order for this to hold for arbitrary δF, it must be that QB |ψ = QB |ψ = 0 .

(4.2.10)

This is the essential condition: physical states must be BRST-invariant. We have assumed that Q†B = QB . There are several ways to see that this must indeed be the case. One is that if Q†B were diﬀerent, it would have to be some other symmetry, and there is no candidate. A better argument is that the ﬁelds cα and bA are like anticommuting versions of the gauge parameter α and Lagrange multiplier BA , and so inherit their reality properties. As an aside, we may also add to the action a term proportional to −1 δB (bA BB M AB ) = −BA BB M AB

(4.2.11)

M AB .

By the above argument, amplitudes between for any constant matrix physical states are unaﬀected. The integral over BA now produces a Gaussian rather than a delta function: these are the Gaussian-averaged gauges, which include for example the covariant α-gauges of gauge theory. There is one more key idea. In order to move around in the space of gauge choices, the BRST charge must remain conserved. Thus it must commute with the change in the Hamiltonian, 0 = [ QB , { QB , bA δF A } ] = Q2B bA δF A − QB bA δF A QB + QB bA δF A QB − bA δF A Q2B = [ Q2B , bA δF A ] .

(4.2.12)

128

4 The string spectrum

In order for this to vanish for general changes of gauge, we need Q2B = 0 .

(4.2.13) Q2B

= constant is That is, the BRST charge is nilpotent; the possibility 2 excluded because QB has ghost number 2. The reader can check that acting twice with the BRST transformation (4.2.6) leaves all ﬁelds invariant. In particular, 1 (4.2.14) δB (δB cα ) = − f αβγ f γδ cβ cδ c = 0 . 2 The product of ghosts is antisymmetric on the indices β, δ, , and the product of structure functions then vanishes by the Jacobi identity. We should mention that we have made two simplifying assumptions about the gauge algebra (4.2.1). The ﬁrst is that the structure constants f αβγ are constants, independent of the ﬁelds, and the second is that the algebra does not have, on the right-hand side, additional terms proportional to the equations of motion. More generally, both of these assumptions break down. In these cases, the BRST formalism as we have described it does not give a nilpotent transformation, and a generalization, the Batalin– Vilkovisky (BV) formalism, is needed. The BV formalism has had various applications in string theory, but we will not have a need for it. The nilpotence of QB has an important consequence. A state of the form QB |χ

(4.2.15)

will be annihilated by QB for any χ and so is physical. However, it is orthogonal to all physical states including itself:

ψ| (QB |χ ) = ( ψ|QB ) |χ = 0

(4.2.16)

if QB |ψ = 0. All physical amplitudes involving such a null state thus vanish. Two physical states that diﬀer by a null state, |ψ = |ψ + QB |χ

(4.2.17)

will have the same inner products with all physical states and are therefore physically equivalent. So, just as for the OCQ, we identify the true physical space with a set of equivalence classes, states diﬀering by a null state being equivalent. This is a natural construction for a nilpotent operator, and is known as the cohomology of QB . Other examples of nilpotent operators are the exterior derivative in diﬀerential geometry and the boundary operator in topology. In cohomology, the term closed is often used for states annihilated by QB and the term exact for states of the form (4.2.15). Thus, our prescription is Hclosed . (4.2.18) HBRST = Hexact

129

4.2 BRST quantization

We will see explicitly in the remainder of the chapter that this space has the expected form in string theory. Essentially, the invariance condition removes one set of unphysical X µ oscillators and one set of ghost oscillators, and the equivalence relation removes the other set of unphysical X µ oscillators and the other set of ghost oscillators. Point-particle example Let us consider an example, the point particle. Expanding out the condensed notation above, the local symmetry is coordinate reparameterization δτ(τ), so the index α just becomes τ and a basis of inﬁnitesimal transformations is δτ1 τ(τ) = δ(τ − τ1 ). These act on the ﬁelds as δτ1 X µ (τ) = −δ(τ − τ1 )∂τ X µ (τ) ,

δτ1 e(τ) = −∂τ [δ(τ − τ1 )e(τ)] . (4.2.19)

Acting with a second transformation and forming the commutator, we have [δτ1 , δτ2 ]X µ (τ)

= − δ(τ − τ1 )∂τ δ(τ − τ2 ) − δ(τ − τ2 )∂τ δ(τ − τ1 ) ∂τ X µ (τ) ≡

dτ3 f τ3τ1 τ2 δτ3 X µ (τ) .

(4.2.20)

From the commutator we have determined the structure function f τ3τ1 τ2 = δ(τ3 − τ1 )∂τ3 δ(τ3 − τ2 ) − δ(τ3 − τ2 )∂τ3 δ(τ3 − τ1 ) .

(4.2.21)

The BRST transformation is then δB X µ δB e δB B δB b δB c

= = = = =

˙µ , icX ˙ , i(ce)

(4.2.22a) (4.2.22b) (4.2.22c) (4.2.22d) (4.2.22e)

0, B , ic˙c .

The gauge e(τ) = 1 is analogous to the unit gauge for the string, using the single coordinate freedom to ﬁx the single component of the tetrad, so F(τ) = 1 − e(τ). The gauge-ﬁxed action is then

S=

dτ

1 −1 ˙ µ ˙ 1 e X Xµ + em2 + iB(e − 1) − e˙bc 2 2

.

(4.2.23)

We will ﬁnd it convenient to integrate B out, thus ﬁxing e = 1. This leaves only the ﬁelds X µ , b, and c, with action

S=

1 ˙µ˙ 1 dτ X Xµ + m2 − ˙bc 2 2

(4.2.24)

130

4 The string spectrum

and BRST transformation ˙µ , δB X µ = icX 1 ˙µ˙ 1 δB b = i − X Xµ + m2 − ˙bc , 2 2 δB c = ic˙c .

(4.2.25a) (4.2.25b) (4.2.25c)

Since B is no longer present, we have used the equation of motion from e to replace B in the transformation law for b. The reader can check that the new transformation (4.2.25) is a symmetry of the action and is nilpotent. This always works when the ﬁelds B A are integrated out, though δB δB b is no longer identically zero but is proportional to the equations of motion. This is perfectly satisfactory: Q2B = 0 holds as an operator equation, which is what we need. The canonical commutators are [pµ , X ν ] = −iη µν ,

{b, c} = 1 ,

(4.2.26)

˙ µ , the i being from the Euclidean signature. The Hamiltonian with pµ = iX is H = 12 (p2 + m2 ), and the Noether procedure gives the BRST operator QB = cH .

(4.2.27)

The structure here is analogous to what we will ﬁnd for the string. The constraint (the missing equation of motion) is H = 0, and the BRST operator is c times this. The ghosts generate a two-state system, so a complete set of states is |k, ↓ , |k, ↑ , where pµ |k, ↓ = k µ |k, ↓ , pµ |k, ↑ = k µ |k, ↑ , b|k, ↓ = 0 , b|k, ↑ = |k, ↓ , c|k, ↓ = |k, ↑ , c|k, ↑ = 0 .

(4.2.28a) (4.2.28b) (4.2.28c)

The action of the BRST operator on these is QB |k, ↓ = 12 (k 2 + m2 )|k, ↑ ,

QB |k, ↑ = 0 .

(4.2.29)

From this it follows that the closed states are |k, ↓ , |k, ↑ ,

k 2 + m2 = 0 , all k µ ,

(4.2.30a) (4.2.30b)

and that the exact states are |k, ↑ ,

k 2 + m2 = 0 .

(4.2.31)

The closed states that are not exact are |k, ↓ , k 2 + m2 = 0 ;

|k, ↑ , k 2 + m2 = 0 .

(4.2.32)

4.3 BRST quantization of the string

131

That is, physical states must satisfy the mass-shell condition, but we have two copies of the expected spectrum. In fact, only states |k, ↓ satisfying the additional condition b|ψ = 0

(4.2.33)

appear in physical amplitudes. The origin of this additional condition is kinematic. For k 2 +m2 = 0, the states |k, ↑ are exact — they are orthogonal to all physical states and their amplitudes must vanish identically. So the amplitudes can only be proportional to δ(k 2 + m2 ). But amplitudes in ﬁeld theory and in string theory, while they may have poles and cuts, never have delta functions (except in D = 2 where the kinematics is special). So they must vanish identically. 4.3

BRST quantization of the string

In string theory, the BRST transformation is ¯ µ, δB X µ = i(c∂ + ˜c∂)X ˜X + T ˜ g) , δB b = i(T X + T g ) , δB ˜b = i(T ¯c . δB c = ic∂c , δB˜c = i˜c∂˜

(4.3.1a) (4.3.1b) (4.3.1c)

The reader can derive this following the point-particle example. To the sum of the Polyakov and ghost actions, add the gauge-ﬁxing term

i d2 σ g 1/2 B ab (δab − gab ) . (4.3.2) 4π The transformation δB bab = Bab becomes (4.3.1) after integrating over Bab and using the gab equation of motion to replace Bab in the transformation. The Weyl ghost is just a Lagrange multiplier, making bab traceless. Again, the reader can check nilpotence up to the equations of motion. The similarity between the string BRST transformation (4.3.1) and the particle case (4.2.25) is evident. Replacing the X µ with a general matter CFT, the BRST transformation of the matter ﬁelds is a conformal transformation with v(z) = c(z), while T m replaces T X in the transformation of b. Noether’s theorem gives the BRST current 1 3 : cT g : + ∂2 c , 2 2 3 = cT m + : bc∂c : + ∂2 c , (4.3.3) 2 and correspondingly for ˜B . The ﬁnal term in the current is a total derivative and does not contribute to the BRST charge; it has been added by hand to make the BRST current a tensor. The OPEs of the BRST jB = cT m +

132

4 The string spectrum

current with the ghost ﬁelds and with a general matter tensor ﬁeld are 3 1 1 + 2 j g (0) + T m+g (0) , (4.3.4a) 3 z z z 1 jB (z)c(0) ∼ c∂c(0) , (4.3.4b) z & h 1% jB (z)Om (0, 0) ∼ 2 cOm (0, 0) + h(∂c)Om (0, 0) + c∂Om (0, 0) . (4.3.4c) z z The simple poles reﬂect the BRST transformations (4.3.1) of these ﬁelds. The BRST operator is jB (z)b(0) ∼

1 QB = 2πi

(dz jB − d¯z ˜B ) .

(4.3.5)

By the usual contour argument, the OPE implies g {QB , bm } = Lm m + Lm .

(4.3.6)

In terms of the ghost modes, QB =

∞

˜m (cn Lm cn L −n + ˜ −n )

n=−∞

+

∞ (m − n) ◦

m,n=−∞

2

◦

(cm cn b−m−n + ˜cm˜cn ˜b−m−n )◦◦ + aB (c0 + ˜c0 ) . (4.3.7)

The ordering constant is aB = ag = −1, as follows from the anticommutator g {QB , b0 } = Lm 0 + L0 .

(4.3.8)

There is an anomaly in the gauge symmetry when cm = 26, so we expect some breakdown in the BRST formalism. The BRST current is still conserved: all of the terms in the current (4.3.3) are analytic for any value of the central charge. However, it is no longer nilpotent, {QB , QB } = 0 only if cm = 26 .

(4.3.9)

The shortest derivation of this uses the Jacobi identity, exercise 4.3. More directly it follows from the OPE jB (z)jB (0) ∼ −

cm − 18 cm − 18 2 cm − 26 3 c∂c(0)− c∂ c(0)− c∂ c(0) . (4.3.10) 2z 3 4z 2 12z

This requires a bit of calculation, being careful about minus signs from anticommutation. The single pole implies that {QB , QB } = 0 when cm = 26.

133

4.3 BRST quantization of the string Note also the OPE

cm − 26 1 1 c(0) + 2 jB (0) + ∂jB (0) , (4.3.11) 4 2z z z which implies that jB is a tensor only when cm = 26. Let us point out some important features. The missing equation of motion is just the vanishing of the generator Tab of the conformal transformations, the part of the local symmetry group not ﬁxed by the gauge choice. This is a general result. When the gauge conditions ﬁx the symmetry completely, one can show that the missing equations of motion are trivial due to the gauge invariance of the action. When they do not, so that there is a residual symmetry group, the missing equations of motion require its generators to vanish. They must vanish in the sense (4.1.2) of matrix elements, or more generally in the BRST sense. Denote the generators of the residual symmetry group by GI . The GI are called constraints; ˜ m . They form an algebra for the string these are Lm and L T (z)jB (0) ∼

[GI , GJ ] = ig KIJ GK .

(4.3.12)

Associated with each generator is a pair of ghosts, bI and cI , where {cI , bJ } = δ IJ ,

{cI , cJ } = {bI , bJ } = 0 .

(4.3.13)

The general form of the BRST operator, as illustrated by the string case (4.3.7), is i K I J QB = cI Gm I − g IJ c c bK 2 1 g I m = c GI + GI , 2

(4.3.14)

where Gm I is the matter part of GI and GgI = −ig KIJ cJ bK Gm I

(4.3.15)

GgI

is the ghost part. The and satisfy the same algebra (4.3.12) as the GI . Using the commutators (4.3.12) and (4.3.13), one ﬁnds 1 1 (4.3.16) Q2B = {QB , QB } = − g KIJ g MKL cI cJ cL bM = 0 . 2 2 The last equality follows from the GGG Jacobi identity, which requires g KIJ g MKL to vanish when antisymmetrized on IJL. We have neglected central charge terms; these need to be checked by hand. To reiterate, a constraint algebra is a world-sheet symmetry algebra that is required to vanish in physical matrix elements. When we go on to generalize the bosonic string theory in volume two, it will be easiest to do so directly in terms of the constraint algebra, and we will write the BRST charge directly in the form (4.3.14).

134

4 The string spectrum BRST cohomology of the string

Let us now look at the BRST cohomology at the lowest levels of the string. The inner product is deﬁned by specifying (αµm )† = αµ−m , (bm )† = b−m , (cm )† = c−m ,

(˜αµm )† = α˜µ−m , (˜bm )† = ˜b−m , (˜cm )† = ˜c−m .

(4.3.17a) (4.3.17b) (4.3.17c)

In particular, the Hermiticity of the BRST charge requires that the ghost ﬁelds be Hermitean as well. The Hermiticity of the ghost zero modes forces the inner products of the ground state to take the form open string: closed string:

(4.3.18a)

0; k|c0 |0; k = (2π)26 δ 26 (k − k ) , 26 26

0; k|˜c0 c0 |0; k = i(2π) δ (k − k ) . (4.3.18b)

Here |0; k denotes the matter ground state times the ghost ground state | ↓ , with momentum k. The c0 and ˜c0 insertions are necessary for a nonzero result. For example, 0; k|0; k = 0; k|(c0 b0 + b0 c0 )|0; k = 0; the last equality follows because b0 annihilates both the bra and the ket. The factor of i is needed in the ghost zero-mode inner product for Hermiticity. Inner products of general states are then obtained, as in the earlier calculation (4.1.13), by use of the commutation relations and the adjoints (4.3.17). We will focus on the open string; the closed string discussion is entirely parallel but requires twice as much writing. As in the point-particle case (4.2.33), we will assert (and later show) that physical states must satisfy the additional condition b0 |ψ = 0 .

(4.3.19)

L0 |ψ = {QB , b0 }|ψ = 0 ,

(4.3.20)

This also implies since QB and b0 both annihilate |ψ . The operator L0 is L0 = α (pµ pµ + m2 ) , where α m2 =

∞ n=1

n Nbn + Ncn +

25

(4.3.21)

Nµn

− 1.

(4.3.22)

µ=0

Thus the L0 condition (4.3.20) determines the mass of the string. BRST invariance, with the extra condition (4.3.19), implies that every string state ˆ the space of states satisfying is on the mass shell. We will denote by H the conditions (4.3.19) and (4.3.20). From the commutators {QB , b0 } = L0 ˆ into itself. and [QB , L0 ] = 0, it follows that QB takes H

4.3 BRST quantization of the string

135

ˆ It The inner product (4.3.18a) is not quite the correct one to use in H. 26 is not well deﬁned: the ghost zero modes give zero, while the δ (k − k ) contains a factor δ(0) because the momentum support is restricted to the ˆ a reduced inner product in which mass shell. Therefore we use in H 0 we simply ignore the X and ghost zero modes. It is this inner product that will be relevant for the probabilistic interpretation. One can check ˆ that QB is still Hermitean with the reduced inner product, in the space H. 0 Note that the mass-shell condition determines k in terms of the spatial momentum k (we focus on the incoming case, k 0 > 0), and that we have used covariant normalization for states. Let us now work out the ﬁrst levels of the D = 26 ﬂat spacetime string. At the lowest level, N = 0, we have |0; k ,

−k 2 = −

1 . α

(4.3.23)

This state is invariant, QB |0; k = 0 ,

(4.3.24)

because every term in QB contains either lowering operators or L0 . This also shows that there are no exact states at this level, so each invariant state corresponds to a cohomology class. These are just the states of the tachyon. The mass-shell condition is the same as found in the light-cone quantization of section 1.3 and from the open string vertex operators in section 3.6. The tachyon mass-squared, determined in a heuristic way in chapter 1, is here ﬁxed by the normal ordering constant in QB , which is determined by the requirement of nilpotence. At the next level, N = 1, there are 26 + 2 states, |ψ1 = (e · α−1 + βb−1 + γc−1 )|0; k ,

−k 2 = 0 ,

(4.3.25)

depending on a 26-vector eµ and two constants, β and γ. The norm of this state is

ψ1 ψ1 = 0; k(e* · α1 + β*b1 + γ*c1 )(e · α−1 + βb−1 + γc−1 )|0; k

= (e* · e + β*γ + γ*β) 0; k0; k . (4.3.26) Going to an orthogonal basis, there are 26 positive-norm states and 2 negative-norm states. The BRST condition is 0 = QB |ψ1 = (2α )1/2 (c−1 k · α1 + c1 k · α−1 )|ψ1

= (2α )1/2 (k · ec−1 + βk · α−1 )|0; k .

(4.3.27)

The terms proportional to c0 sum to zero by the mass-shell condition, and are omitted. An invariant state therefore satisﬁes k · e = β = 0. There are

136

4 The string spectrum

26 linearly independent states remaining, of which 24 have positive norm and 2 have zero norm, being orthogonal to all physical states including themselves. The zero-norm invariant states are created by c−1 and k · α−1 . A general |χ is of the same form (4.3.25) with constants eµ , β , γ , so the general BRST-exact state at this level is QB |χ = (2α )1/2 (k · e c−1 + β k · α−1 )|0; k .

(4.3.28)

Thus the ghost state c−1 |0; k is BRST-exact, while the polarization is transverse with the equivalence relation eµ ∼ = eµ + (2α )1/2 β kµ . This leaves the 24 positive-norm states expected for a massless vector particle, in agreement with the light-cone quantization and with the OCQ at A = −1. This pattern is general: there are two extra positive-norm and two extra negative-norm families of oscillators, as compared to the lightcone quantization. The physical state condition eliminates two of these and leaves two combinations with vanishing inner products. These null oscillators are BRST-exact and are removed by the equivalence relation. An aside: the states b−1 |0; k and c−1 |0; k can be identiﬁed with the two Faddeev–Popov ghosts that appear in the BRST-invariant quantization of a massless vector ﬁeld in ﬁeld theory. The world-sheet BRST operator acts on these states in the same way as the corresponding spacetime BRST operator in gauge ﬁeld theory. Acting on the full string Hilbert space, the open string BRST operator is then some inﬁnite-dimensional generalization of the spacetime gauge theory BRST invariance, while the closed string BRST operator is a generalization of the spacetime general coordinate BRST invariance, in the free limit. The generalization to the closed string is straightforward. We restrict ˆ of states satisfying1 attention to the space H b0 |ψ = ˜b0 |ψ = 0 ,

(4.3.29)

˜ 0 |ψ = 0 . L0 |ψ = L

(4.3.30)

implying also

In the closed string, L0 =

1

α 2 (p + m2 ) , 4

˜ 0 = α (p2 + m ˜ 2) , L 4

(4.3.31)

S. Weinberg has pointed out that the simple argument given at the end of section 4.2 would only lead to the weaker condition (b0 + ˜ b0 )|ψ = 0. Nevertheless, we shall see from the detailed form of string amplitudes in chapter 9 that there is a projection onto states satisfying conditions (4.3.29).

137

4.4 The no-ghost theorem where

∞ 25 α 2 m = n Nbn + Ncn + Nµn 4 n=1 µ=0

∞ 25 α 2 ˜ bn + N ˜ µn ˜ cn + ˜ = N n N m 4 n=1 µ=0

− 1,

(4.3.32a)

− 1.

(4.3.32b)

Repeating the earlier exercise, we ﬁnd at m2 = −4/α the tachyon, and at m2 = 0 the 24 × 24 states of the graviton, dilaton, and antisymmetric tensor.

4.4

The no-ghost theorem

In this section we prove that the BRST cohomology of the string has a positive inner product and is isomorphic to the light-cone and OCQ spectra. We are identifying the BRST cohomology as the physical Hilbert space. We will also need to verify, in our study of string amplitudes, that the amplitudes are well deﬁned on the cohomology (that is, that equivalent states have equal amplitudes) and that the S-matrix is unitary within the space of physical states. We will work in the general framework described at the end of chapter 3. That is, the world-sheet theory consists of d free X µ ﬁelds including µ = 0, plus some compact unitary CFT K of central charge 26 − d, plus the ghosts. The Virasoro generators are a sum K g Lm = LX m + Lm + Lm .

(4.4.1)

By compact we mean that LK 0 has a discrete spectrum. For example, in the case that K corresponds to strings on a compact manifold, the term α p2 in L0 is replaced by the Laplacian −α ∇2 , which has a discrete spectrum on a compact space. The general state is labeled |N, I; k ,

˜ I; k , |N, N,

(4.4.2)

˜ refer to both in the open and closed strings respectively, where N (and N) the d-dimensional and ghost oscillators, k is the d-momentum, and I labels the states of the compact CFT with given boundary conditions. The b0 condition (4.3.19) or (4.3.29) is imposed as above, implying the mass-shell

138

4 The string spectrum

condition −

d−1

kµ k µ = m2 ,

µ=0

2

αm =

∞

(4.4.3a)

n Nbn + Ncn +

n=1

d−1

+ LK 0 − 1 ,

Nµn

(4.4.3b)

µ=0

for the open string and −

d−1 µ=0

˜2 , kµ k µ = m2 = m

(4.4.4a)

∞ d−1 α 2 n Nbn + Ncn + Nµn m = 4 n=1 µ=0

∞ d−1 α 2 ˜ bn + N ˜ µn ˜ cn + ˜ = N n N m 4 n=1 µ=0

+ LK 0 − 1 ,

(4.4.4b)

˜K + L 0 − 1 ,

(4.4.4c)

for the closed string. That is, the contributions of the d ≤ µ ≤ 25 oscillators ˜K are replaced by the eigenvalue of LK 0 or L0 from the compact CFT. The only information we use about the compact CFT is that it is conformally invariant with the appropriate central charge, so there is a nilpotent BRST operator, and that it has a positive inner product. The basis I can be taken to be orthonormal, so the reduced inner products are

0, I; k0, I ; k = 0, 0, I; k0, 0, I ; k = 2k 0 (2π)d−1 δ d−1 (k − k )δI,I . (4.4.5) Now let us see what we expect for the physical Hilbert space. Deﬁne ˆ that have the transverse Hilbert space H⊥ to consist of those states in H 0 1 no longitudinal (X , X , b, or c) excitations. Since these oscillators are the source of the indeﬁnite inner product, H⊥ has a positive inner product. Light-cone gauge-ﬁxing eliminates the longitudinal oscillators directly — the light-cone Hilbert space is isomorphic to H⊥ , as one sees explicitly in the ﬂat-spacetime case from chapter 1. We will show, in the general case, that the BRST cohomology is isomorphic to H⊥ . That is, it has the same number of states at each mass level, and has a positive inner product. This is the no-ghost theorem. Proof The proof has two parts. The ﬁrst is to ﬁnd the cohomology of a simpliﬁed BRST operator Q1 , which is quadratic in the oscillators, and the second is to show that the cohomology of the full QB is identical to that of Q1 .

139

4.4 The no-ghost theorem Deﬁne the light-cone oscillators −1/2 0 α± (αm ± α1m ) , m =2

(4.4.6)

which satisfy − [α+ m , αn ] = −mδm,−n ,

+ − − [α+ m , αn ] = [αm , αn ] = 0 .

(4.4.7)

We will use extensively the quantum number N lc =

∞ 1 m=−∞ m=0

m

◦ ◦

− α+ −m αm

◦ ◦

(4.4.8)

.

The number N lc counts the number of − excitations minus the number of + excitations; it is not a Lorentz generator because the center-ofmass piece has been omitted. We choose a Lorentz frame in which the momentum component k + is nonzero. Now decompose the BRST generator using the quantum number N lc : QB = Q1 + Q0 + Q−1 ,

(4.4.9)

where Qj changes N lc by j units, [N lc , Qj ] = jQj .

(4.4.10)

Also, each of the Qj increases the ghost number N g by one unit: [N g , Qj ] = Qj . Expanding out Q2B = 0 gives

(4.4.11)

Q21 + {Q1 , Q0 } + {Q1 , Q−1 }+Q20 + {Q0 , Q−1 } + Q2−1 = 0 . (4.4.12)

Each group in parentheses has a diﬀerent N lc and so must vanish separately. In particular, Q1 itself is nilpotent, and so has a cohomology. Explicitly, Q1 = −(2α )1/2 k +

∞

α− −m cm .

(4.4.13)

m=−∞ m=0

For m < 0 this destroys a + mode and creates a c, and for m > 0 it creates a − mode and destroys a b. One can ﬁnd the cohomology directly by considering the action of Q1 in the occupation basis. We will leave this as an exercise, and instead use a standard strategy which will also be useful in generalizing to QB . Deﬁne ∞ 1 α+ b R= 1/2 + (2α ) k m=−∞ −m m m=0

(4.4.14)

140

4 The string spectrum

and S ≡ {Q1 , R} = =

∞ ' m=1 ∞

− − + mb−m cm + mc−m bm − α+ −m αm − α−m αm

m(Nbm + Ncm + Nm+ + Nm− ) .

(

(4.4.15)

m=1

The normal ordering constant is determined by noting that Q1 and R both annihilate the ground state. Note that Q1 commutes with S . We can then calculate the cohomology within each eigenspace of S , and the full cohomology is the union of the results. If |ψ is Q1 -invariant with S |ψ = s|ψ , then for nonzero s 1 1 |ψ = {Q1 , R}|ψ = Q1 R|ψ , s s

(4.4.16)

and so |ψ is actually Q1 -exact. Therefore the Q1 cohomology can be nonzero only at s = 0. By the deﬁnition (4.4.15) of S , the s = 0 states have no longitudinal excitations — the s = 0 space is just H⊥ . The operator Q1 annihilates all states in H⊥ , so they are all Q1 -closed and there are no Q1 -exact states in this space. Therefore the cohomology is H⊥ itself. We have proven the no-ghost theorem, but for the operator Q1 , not QB . The proof had two steps, the ﬁrst being to show that the cohomology could only come from s = 0 states (the kernel of S ), and the second to show that s = 0 states were Q1 -invariant. It is useful to prove the second step in a more abstract way, using the property that all s = 0 states have the same ghost number, in this case − 12 . Suppose S |ψ = 0. Since S and Q1 commute we have 0 = Q1 S |ψ = S Q1 |ψ .

(4.4.17)

The state |ψ has ghost number − 12 , so Q1 |ψ has ghost number + 12 . Since S is invertible at this ghost number, it must be that Q1 |ψ = 0 as we wished to show. It remains to show that the cohomology of QB is the same as the cohomology of Q1 . The idea here is to use in place of S the operator S + U ≡ {QB , R} .

(4.4.18)

Now, U = {Q0 + Q−1 , R} lowers N lc by one or two units. In terms of N lc y, S is diagonal and U is lower triangular. By general properties of lower triangular matrices, the kernel of S + U can be no larger than the kernel of its diagonal part S . In fact they are isomorphic: if |ψ0 is annihilated by S , then |ψ = (1 − S −1 U + S −1 US −1 U − . . .)|ψ0

(4.4.19)

4.4 The no-ghost theorem

141

is annihilated by S + U. The factors of S −1 make sense because they always act on states of N lc < 0, where S is invertible. For the same reason, S + U is invertible except at ghost number − 12 . Eqs. (4.4.16) and (4.4.17) can now be applied to QB , with S + U replacing S . They imply that the QB cohomology is isomorphic to the kernel of S + U, which is isomorphic to the kernel of S , which is isomorphic to the cohomology of Q1 , as we wished to show. We must still check that the inner product is positive. All terms after the ﬁrst on the right-hand side of eq. (4.4.19) have strictly negative N lc . By the commutation relations, the inner product is nonzero only between states whose N lc adds to zero. Then for two states (4.4.19) in the kernel of S + U,

ψψ = ψ0 ψ0 .

(4.4.20)

The positivity of the inner product on the kernel of S + U then follows from that for the kernel of S , and we are done. After adding the tilded operators to QB , N lc , R, etc., the closed string proof is identical. BRST–OCQ equivalence We now prove the equivalence HOCQ = HBRST = Hlight-cone .

(4.4.21)

To a state |ψ in the matter Hilbert space, associate the state |ψ, ↓

(4.4.22)

from the full matter plus ghost theory. Again the ghost vacuum | ↓ is annihilated by all of the ghost lowering operators, bn for n ≥ 0 and cn for n > 0. Acting with QB , QB |ψ, ↓ =

∞

c−n (Lm n − δn,0 )|ψ, ↓ = 0 .

(4.4.23)

n=0

All terms in QB that contain ghost lowering operators drop out, and the constant in the n = 0 term is known from the mode expansion of QB , eq. (4.3.7). Each OCQ physical state thus maps to a BRST-closed state. The ordering constant A = −1 arises here from the L0 eigenvalue of the ghost vacuum. To establish the equivalence (4.4.21), we need to show more. First, we need to show that we have a well-deﬁned map of equivalence classes: that if ψ and ψ are equivalent OCQ physical states, they map into the same BRST class: |ψ, ↓ − |ψ , ↓

(4.4.24)

142

4 The string spectrum

must be BRST-exact. By (4.4.23), this state is BRST-closed. Further, since |ψ − |ψ is OCQ null, the state (4.4.24) has zero norm. From the BRST no-ghost theorem, the inner product on the cohomology is positive, so a zero-norm closed state is BRST-exact, as we needed to show. To conclude that we have an isomorphism, we need to show that the map is one-to-one and onto. One-to-one means that OCQ physical states ψ and ψ that map into the same BRST class must be in the same OCQ class — if |ψ, ↓ − |ψ , ↓ = QB |χ ,

(4.4.25)

then |ψ − |ψ must be OCQ null. To see this, expand |χ =

∞

b−n |χn , ↓ + . . . ;

(4.4.26)

n=1

|χ has ghost number − 32 , so the ellipsis stands for terms with at least one c and two b excitations. Insert this into the form (4.4.25) and keep on both sides only terms with the ghost ground state. This gives |ψ, ↓ − |ψ , ↓ = =

∞

cm Lm −m b−n |χn , ↓

m,n=1 ∞

Lm −n |χn , ↓ .

(4.4.27)

n=1

Terms with ghost excitations must vanish separately, and so have been m omitted. Thus, |ψ − |ψ = ∞ n=1 L−n |χn is OCQ null, and the map is one-to-one. Finally, we must show that the map is onto, that every QB class contains at least one state of the form (4.4.22). In fact, the speciﬁc representatives (4.4.19), the states annihilated by S + U, are of this form. To see this, consider the quantum number N = 2N − + Nb + Nc , involving the total numbers of −, b, and c excitations. The operator R has N = −1: terms in R with m > 0 reduce Nc by one unit and terms with m < 0 reduce N − by one unit and increase Nb by one unit. Examining Q0 + Q−1 , one ﬁnds various terms with N = 1, but no greater. So U = {R, Q0 + Q−1 } cannot increase N . Examining the state (4.4.19), and noting that S and |ψ0 have N = 0, we see that all terms on the right-hand side have N ≤ 0. By deﬁnition N is nonnegative, so it must be that N |ψ = 0. This implies no −, b, or c excitations, and so this state is of the form (4.4.22). Thus the equivalence (4.4.21) is shown. The full power of the BRST method is needed for understanding the general structure of string amplitudes. However, for many practical purposes it is a great simpliﬁcation to work with the special states of the

Exercises

143

form |ψ, ↓ , and so it is useful to know that every BRST class contains at least one state in which the ghost modes are unexcited. We will call these OCQ-type states. The OCQ physical state condition (4.1.5) requires that the matter state be a highest weight state with L0 = 1. The corresponding vertex operator is a weight-1 tensor ﬁeld Vm constructed from the matter ﬁelds. Including the ghost state | ↓ , the full vertex operator is cVm . For the closed string the full vertex operator is ˜ccVm with Vm a (1, 1) tensor. The matter part of the vertex operator is the same as found in the Polyakov formalism in section 3.6, while the ghost part has a simple interpretation that we will encounter in the next chapter. Eq. (4.4.19) deﬁnes an OCQ-physical state in each cohomology class. For the special case of ﬂat spacetime this state can be constructed more explicitly by using the Del Giudice–Di Vecchia–Fubini (DDF) operators. To explain these we will need to develop some more vertex operator technology, so this is deferred to chapter 8. Exercises 4.1 Extend the OCQ, for A = −1 and general D, to the second excited level of the open string. Verify the assertions made in the text about extra positive- and negative-norm states. 4.2 (a) In the OCQ, show that the state Lm −1 |χ is physical (and therefore null) if |χ is a highest weight state with Lm 0 = 0. 3 m m + L L )|χ

is physical and null if |χ is a (b) Show that the state (Lm −2 2 −1 −1 = −1. highest weight state with Lm 0 4.3 By writing out terms, demonstrate the Jacobi identity { [QB , Lm ] , bn } − { [Lm , bn ] , QB } − [ {bn , QB } , Lm ] = 0 . This is the sum of cyclic permutations, with commutators or anticommutators as appropriate, and minus signs from anticommutation. Use the Jacobi identity and the (anti)commutators (2.6.24) and (4.3.6) of bn with Ln and QB to show that { [QB , Lm ] , bn } vanishes when the total central charge is zero. This implies that [QB , Lm ] contains no c modes. However, this commutator has N g = 1, and so it must vanish: the charge QB is conformally invariant. Now use the QB QB bn Jacobi identity in the same way to show that QB is nilpotent when the total central charge is zero. 4.4 In more general situations, one encounters graded Lie algebras, with bosonic generators (fermion number FI even) and fermionic generators (FI odd). The algebra of constraints is GI GJ − (−1)FI FJ GJ GI = ig KIJ GK .

144

4 The string spectrum

Construct a nilpotent BRST operator. You will need bosonic ghosts for the fermionic constraints. 4.5 (a) Carry out the BRST quantization for the ﬁrst two levels of the closed string explicitly. (b) Carry out the BRST quantization for the third level of the open string, m2 = 1/α . − 4.6 (a) Consider the operator Q = α− −1 c1 + α1 c−1 , obtained by truncating Q1 to the m = ±1 oscillators only. Calculate the cohomology of Q directly, by considering its action on a general linear combination of the states −

N N (α− |0 . (b−1 )Nb (c−1 )Nc (α+ −1 ) −1 ) +

(b) Generalize this to the full Q1 .

5 The string S-matrix

In chapter 3 we wrote the string S-matrix as a path integral over compact two-dimensional surfaces with vertex operators. In this chapter we will reduce the path integral to gauge-ﬁxed form. 5.1

The circle and the torus

We would like to identify a gauge slice, a choice of one conﬁguration from each (diﬀ×Weyl)-equivalent set as in ﬁgure 3.7. Locally we did this by ﬁxing the metric, but globally there is a small mismatch between the space of metrics and the world-sheet gauge group. Once again the point particle is a good illustration. We want to evaluate the Euclidean path integral

1 [de dX] exp − 2

−1 ˙ µ ˙

2

dτ (e X Xµ + em ) .

(5.1.1)

Consider a path forming a closed loop in spacetime, so the topology is a circle. The parameter τ can be taken to run from 0 to 1 with the endpoints identiﬁed. That is, X µ (τ) and e(τ) are periodic on 0 ≤ τ ≤ 1. The tetrad e(τ) has one component and there is one local symmetry, the choice of parameter, so as for the string there is just enough local symmetry to ﬁx the tetrad completely. The tetrad transforms as e dτ = edτ. The gauge choice e = 1 thus gives a diﬀerential equation for τ (τ), ∂τ = e(τ) . ∂τ

(5.1.2)

Integrating this with the boundary condition τ (0) = 0 determines τ (τ) =

τ

dτ e(τ ) .

0

145

(5.1.3)

146

5 The string S-matrix

The complication is that in general τ (1) = 1, so the periodicity is not preserved. In fact,

1

τ (1) =

dτ e(τ) = l

(5.1.4)

0

is the invariant length of the circle. So we cannot simultaneously set e = 1 and keep the coordinate region ﬁxed. We can hold the coordinate region ﬁxed and set e to the constant value e = l, or set e = 1 and let the coordinate region vary: e = l ,

0≤τ≤1,

(5.1.5a)

e = 1 ,

0≤τ≤l .

(5.1.5b)

or

In either case, after ﬁxing the gauge invariance we are left with an ordinary integral over l. In other words, not all tetrads on the circle are diﬀ-equivalent. There is a one-parameter family of inequivalent tetrads, parameterized by l. Both descriptions (5.1.5) will have analogs in the string. In practice, we will deﬁne the path integral using a description like (5.1.5a), where the ﬁelds are functions on a ﬁxed coordinate region, and then convert to a description like (5.1.5b) where the metric is ﬁxed and the moduli are encoded in the coordinate range. There is a second complication with the gauge-ﬁxing. The condition e = constant is preserved by the rigid translation τ → τ + v mod 1 .

(5.1.6)

That is, we have not said where on the circle to put the origin of the coordinate system. Fixing the metric thus leaves a small part of the local symmetry unﬁxed. So there is a small mismatch in both directions, metrics that are not gauge-equivalent to one another and gauge transformations that are not ﬁxed by the choice of metric. Moving on to the string we take as an example the torus, where the same complications arise. Start with the coordinate region 0 ≤ σ 1 ≤ 2π ,

0 ≤ σ 2 ≤ 2π ,

(5.1.7)

with X µ (σ 1 , σ 2 ) and gab (σ1 , σ2 ) periodic in both directions. Equivalently we can think of this as the σ plane with the identiﬁcation of points, (5.1.8) (σ 1 , σ 2 ) ∼ = (σ 1 , σ 2 ) + 2π(m, n) for integer m and n. To what extent is the ﬁeld space diﬀ×Weyl redundant? The theorem is that it is not possible to bring a general metric to unit form by a

147

5.1 The circle and the torus

2 π (τ +1) B A w 2π

0

¯ The upper and lower Fig. 5.1. Torus with modulus τ, in gauge ds2 = dwdw. edges are identiﬁed, as are the right- and left-hand edges. Closed curves A and B are marked for later reference.

diﬀ×Weyl transformation that leaves invariant the periodicity (5.1.7), but it is possible to bring it to the form ds2 = |dσ 1 + τdσ 2 |2

(5.1.9)

where τ is a complex constant. For τ = i this would be the unit metric δab . To see this, repeat the steps used in the local discussion in section 3.3. We can ﬁrst make the metric ﬂat by a Weyl transformation satisfying 2∇2 ω = R. By expanding in eigenmodes of ∇2 , one sees that with the periodic boundary conditions this has a solution up to addition unique 2 1/2 of a constant to ω. It is important here that d σ g R, which is 4π times the Euler number, vanishes for the torus. A transformation to new ˜ a then brings the metric to unit form. However, as in the coordinates σ point-particle case, there is no guarantee that this respects the original periodicity. Rather, we may now have ˜ a + 2π(mua + nv a ) ˜a ∼ σ = σ ua

(5.1.10)

va.

and By rotating and rescaling the coordiwith general translations nate system, accompanied by a shift in ω to keep the metric normalized, we can always set u = (1, 0). This leaves two parameters, the components ¯ and the periodicity is ˜ 1 + i˜ σ 2 , the metric is dwdw of v. Deﬁning w = σ w∼ = w + 2π(m + nτ) ,

(5.1.11)

where τ = v 1 + iv 2 . The torus is a parallelogram in the w-plane with periodic boundary conditions, as shown in ﬁgure 5.1. Alternatively, deﬁne σ a by w = σ 1 + τσ 2 . The original periodicity (5.1.8) is preserved but the metric now takes the more general form (5.1.9). The integration over metrics reduces to two ordinary integrals, over the real and imaginary parts of τ. The metric (5.1.9) is invariant under complex

148

5 The string S-matrix

conjugation of τ and degenerate for τ real, so we can restrict attention to Im τ > 0. As in the case (5.1.5) of the circle, we can put these parameters either in the metric (5.1.9) or the periodicity (5.1.11). The parameter τ is known as a Teichm¨uller parameter or more commonly a modulus. A diﬀerence from the point-particle case is that there is no simple invariant expression for τ analogous to the invariant length of the circle. There is some additional redundancy, which does not have an analog in the point-particle case. The value τ + 1 generates the same set of identiﬁcations (5.1.11) as τ, replacing (m, n) → (m−n, n). So also does −1/τ, deﬁning w = τw and replacing (m, n) → (n, −m). Repeated application of these two transformations, T : τ = τ + 1 ,

S : τ = −1/τ ,

(5.1.12)

generates τ =

aτ + b cτ + d

(5.1.13)

for all integer a, b, c, d such that ad − bc = 1. We can also think about this as follows. The transformation

σ1 σ2

=

d b c a

σ 1 σ 2

(5.1.14)

takes the metric (5.1.9) for σ into a metric of the same form in σ but with the modulus τ . This is a diﬀeomorphism of the torus. It is one-to-one as a consequence of ad − bc = 1 and it preserves the periodicity (5.1.8). However, it cannot be obtained from the identity by successive inﬁnitesimal transformations — it is a so-called large coordinate transformation. The curve A in the coordinate σ maps to a curve in the σ coordinate that runs a times in the A direction and −c times in the B direction. These large coordinate transformations form the group S L(2, Z), integer-valued 2 × 2 matrices with unit determinant. The group on the τ-plane is S L(2, Z)/Z2 = P S L(2, Z), because τ is unchanged if all the signs of a, b, c, d are reversed. The group of transformations (5.1.14) is known as the modular group. Using the modular transformations (5.1.13), it can be shown that every τ is equivalent to exactly one point in the region F0 shown in ﬁgure 5.2, 1 1 ≤ Re τ ≤ , |τ| ≥ 1 , (5.1.15) 2 2 except on the boundaries which are identiﬁed as shown in the ﬁgure. This is called a fundamental region for the upper half-plane mod P S L(2, Z). Because of the identiﬁcations one should think of F0 as being rolled up, and open only at Im τ → ∞. The fundamental region F0 is one representation of the moduli space of (diﬀ×Weyl)-inequivalent metrics. −

149

5.1 The circle and the torus

I

I'

i II

II'

III

III'

τ 1 − 2

1 −− 2

Fig. 5.2. The standard fundamental region F0 for the moduli space of the torus, shaded. The lines I and I are identiﬁed, as are the arcs II and II . A diﬀerent fundamental region, mapped into F0 by the modular transformation S, is bounded by II, II , III, and III .

There is a further complication as there was for the point particle. Requiring the metric to take the form (5.1.9) with τ in a given fundamental region does not ﬁx all of the diﬀ×Weyl invariance. The metric and periodicity are left invariant by rigid translations, σa → σa + va ,

(5.1.16)

so this two-parameter subgroup of diﬀ×Weyl is not ﬁxed. In addition, the discrete transformation σ a → −σ a leaves the metric invariant, and in the unoriented case also (σ 1 , σ 2 ) → (−σ 1 , σ 2 ) with τ → −¯τ. Thus there are again two kinds of mismatch between the metric degrees of freedom and the local symmetries: parameters in the metric that cannot be removed by the symmetries, and symmetries that are not ﬁxed by the choice of metric. The unﬁxed symmetries are known as the conformal Killing group (CKG). When there are enough vertex operators in the amplitude, we can fully ﬁx the gauge invariance by ﬁxing some of the vertex operator positions. On the torus with n vertex operators, the invariance (5.1.16) can be used

150

5 The string S-matrix

to ﬁx one vertex operator position, leaving an integral over τ and the n − 1 other positions. The Z2 from σ a → −σ a can be ﬁxed by restricting a second vertex operator to half the torus. Actually, for ﬁnite overcounting it is often more convenient to leave some symmetry unﬁxed and divide by an appropriate factor. Also, if there are too few vertex operators, some conformal Killing invariance remains unﬁxed and we must allow explicitly for the overcounting. A note to the reader In the remainder of this chapter we discuss moduli in more generality, and obtain the measure for the gauge-ﬁxed integral. The principal goal is the gauge-ﬁxed expression (5.3.9) for the string S-matrix. In fact, most of the interesting string physics can be seen in the tree-level and one-loop amplitudes. For these the measure can be obtained by shortcuts. For the tree-level amplitudes there are no metric moduli but some vertex operator positions must be ﬁxed. The measure can be deduced as explained below eq. (6.4.4). For one-loop amplitudes the measure can be understood in a fairly intuitive way, by analogy to the discussion of the point particle measure below eq. (7.3.9). Thus it is possible for the reader to skip the remainder of this chapter (or to skim as far as eq. (5.3.9)), and over all sections dealing with the Faddeev–Popov ghosts. 5.2

Moduli and Riemann surfaces

Now let us repeat the preceding discussion in a more general and abstract way. We start with the integral over all metrics on some given topology r. Call the space of metrics Gr . For closed oriented surfaces we can label r simply by the number g of handles, also known as the genus. After taking into account the diﬀ×Weyl redundancy, we are left with the moduli space Mr =

Gr . (diﬀ × Weyl)r

(5.2.1)

As in the case of the torus, this space is parameterized by a ﬁnite number of moduli. For the torus, M1 is the upper half-plane mod P S L(2, Z), or equivalently any fundamental region, say F0 . There may also be a subgroup of diﬀ×Weyl, the CKG, that leaves the metric invariant. When there are vertex operators in the path integral it is useful to treat their positions on the same footing as the moduli from the metric, referring to all as moduli. We specify the metric moduli when we need to make the distinction. One way to deal with the CKG, which is applicable if there are vertex operators in the path integral, is to specify the gauge further by ﬁxing the coordinates of some vertex operators. The Polyakov

151

5.2 Moduli and Riemann surfaces

path integral includes an integral over Gr and integrals of the n vertex operator positions over the world-sheet M. The moduli space at topology r with n vertex operators is then Mr,n =

Gr × Mn . (diﬀ × Weyl)r

(5.2.2)

As we have seen for the torus, diﬀr is not in general connected. Picking out the connected component diﬀr0 that contains the identity, the quotient diﬀ r diﬀ r0

(5.2.3)

is the modular group. It is interesting to study the diﬀ×Weyl redundancy of the metric at the inﬁnitesimal level. That is, we are looking for small variations of the metric that are not equivalent to diﬀ×Weyl transformations and so correspond to changes in the moduli. We are also looking for small diﬀ×Weyl transformations that do not change the metric; these would be inﬁnitesimal elements of the CKG, called conformal Killing vectors (CKVs). This will give further insight into the origin of the moduli and the CKG. An inﬁnitesimal diﬀ×Weyl transformation changes the metric by δgab = −2(P1 δσ)ab + (2δω − ∇ · δσ)gab ,

(5.2.4)

where P1 is the traceless symmetric linear combination of derivatives (3.3.17). Moduli correspond to variations δ gab of the metric that are orthogonal to all variations (5.2.4):

0=

=

d2 σ g 1/2 δ gab −2(P1 δσ)ab + (2δω − ∇ · δσ)g ab

d2 σ g 1/2 −2(P1T δ g)a δσ a + δ gab g ab (2δω − ∇ · δσ) . (5.2.5)

The transpose (P1T u)a = −∇b uab is deﬁned as in exercise 3.2. In order for the overlap (5.2.5) to vanish for general δω and δσ we need g ab δ gab = 0 , (P1T δ g)a = 0 .

(5.2.6a) (5.2.6b)

The ﬁrst condition requires that δ gab be traceless, and it is on traceless symmetric tensors that P1T acts. For each solution of these equations there will be a modulus. CKVs are transformations (5.2.4) such that δgab = 0. The trace of this equation ﬁxes δω uniquely, leaving the conformal Killing equation (P1 δσ)ab = 0 .

(5.2.7)

152

5 The string S-matrix

Eqs. (5.2.6) and (5.2.7) become simple for variations around conformal gauge, ∂¯z δ gzz = ∂z δ g¯z¯z = 0, ∂¯z δz = ∂z δ¯z = 0 ,

(5.2.8a) (5.2.8b)

so variations of the moduli correspond to holomorphic quadratic diﬀerentials and CKVs to holomorphic vector ﬁelds. On the torus, the only holomorphic doubly periodic functions are the constants, so there are two real moduli and two real CKVs, in agreement with the discussion in the previous section. The metric moduli correspond to the kernel (5.2.6) of P1T and the CKVs to the kernel (5.2.7) of P1 . The Riemann–Roch theorem relates the number of metric moduli µ = dim ker P1T and the number of CKVs κ = dim ker P1 to the Euler number χ, µ − κ = −3χ .

(5.2.9)

We will be able to derive this later in the chapter. For a closed oriented surface −3χ = 6g − 6. This counting refers to real moduli, separating complex moduli such as τ into their real and imaginary parts. Further, κ vanishes for χ < 0 and µ vanishes for χ > 0. To show this, we ﬁrst cite the fact that it is always possible by a Weyl transformation to ﬁnd a metric for which the scalar curvature R is constant. The sign of R is then the same as that of χ. Now, P1T P1 = − 12 ∇2 − 14 R, so

d2 σ g 1/2 (P1 δσ)ab (P1 δσ)ab =

=

d2 σ g 1/2 δσa (P1T P1 δσ)a

2

d σg

1/2

R 1 ∇a δσb ∇a δσ b − δσa δσ a . 2 4 (5.2.10)

For negative χ the right-hand side is strictly positive, so P1 δσ cannot vanish. A similar argument shows that P1T δ g cannot vanish for positive χ. Combining these results we have χ > 0 : κ = 3χ , µ = 0 , χ < 0 : κ = 0 , µ = −3χ .

(5.2.11a) (5.2.11b)

Riemann surfaces The familiar way to describe moduli space would be to choose one metric from each equivalence class. This would be a family of metrics gˆ ab (t; σ), depending on the moduli tk . For the torus, eq. (5.1.9) gives such a slice. It is often convenient to use an alternative description of the

5.2 Moduli and Riemann surfaces

153

¯ is ﬁxed and the moduli are encoded form (5.1.11), where the metric dwdw in the coordinate region. We can formalize this idea as follows. Let us ﬁrst recall how diﬀerentiable manifolds are deﬁned. One covers the manifold with a set of overlapping patches, with coordinates σma in the mth patch, with a running from 1 to the dimension of the manifold. When patches m and n overlap, the coordinates in the two patches are related by a σma = fmn (σn ) ,

(5.2.12)

where the transition functions fmn are required to be diﬀerentiable a given number of times. For a Riemannian manifold a metric gm,ab (σm ) is also given in each patch, with the usual tensor transformation law relating the values in the overlaps. For a complex manifold, there are complex coordinates zma in each patch, where now a runs from 1 to half the dimension of the manifold. The transition functions are required to be holomorphic, a (zn ) . zma = fmn

(5.2.13)

One can now deﬁne holomorphic functions on the manifold, since the holomorphicity will not depend on which coordinates zma are used. Just as two diﬀerentiable manifolds are equivalent if there is a one-to-one diﬀerentiable map between them, two complex manifolds are equivalent if there is a one-to-one holomorphic map between them. In particular, making a holomorphic change of coordinates within each patch gives an equivalent surface. In the two-dimensional case (one complex coordinate), a complex manifold is known as a Riemann surface. In this case there is a one-to-one correspondence Riemann surfaces ↔ Riemannian manifolds mod Weyl .

(5.2.14)

On the right we put only ‘mod Weyl’ because ‘mod diﬀ’ is already implicit in the deﬁnition of a Riemannian manifold. To see this isomorphism, start with a Riemannian manifold. We know from our discussion of conformal gauge that we can ﬁnd in each coordinate patch a coordinate zm such that ds2 ∝ dzm d¯zm .

(5.2.15)

In neighboring patches the coordinate need not be the same, but because ds2 ∝ dzm d¯zm ∝ dzn d¯zn , the transition functions will be holomorphic.1 This is the map from Riemannian manifolds to complex manifolds in two real dimensions. For the inverse map, one can take the metric dzm d¯zm in 1

To be precise, an antiholomorphic transition function is also possible. For the oriented string we forbid this by choosing a standard orientation; for the unoriented string it is allowed. The term Riemann surface in the strict sense refers to an oriented manifold without boundary.

154

5 The string S-matrix

the mth patch and smooth it in the overlaps (this is a standard construction, the partition of unity) to produce a Riemannian manifold. The two characterizations of a Riemann surface are thus equivalent. The description of the torus in terms of the complex coordinate w of the parallelogram (ﬁgure 5.1) illustrates the idea of a complex manifold. Imagine taking a single coordinate patch which is a little larger than the fundamental parallelogram in ﬁgure 5.1. The periodicity conditions w ∼ = w + 2π ∼ = w + 2πτ

(5.2.16)

are then the transition functions on the overlap between the opposite edges of the patch. The surface is deﬁned by these transition functions. To deﬁne the original path integral over metrics, it is simplest to start by regarding the metric as a function of ﬁxed coordinates such as the square (5.1.7), or at least on a ﬁxed set of coordinate patches with ﬁxed transition functions. To study the quantum ﬁeld theory on a given surface, it then is easiest to work with the unit metric with moduli-dependent transition functions, as in the parallelogram (5.1.11). One can also think of a Riemann surface as follows. Deﬁning the world-sheet as a union of patches, we can use the diﬀ×Weyl freedom to reach the metric dzd¯z in each. The gauge choices on overlapping patches can then diﬀer only by diﬀ×Weyl invariances of this metric, which as we have discussed are the conformal transformations. A Riemann surface is therefore the natural place for a CFT to live, because the ﬁelds in a CFT have deﬁnite transformation properties under conformal transformations. 5.3

The measure for moduli

We now revisit the gauge-ﬁxing of the Polyakov path integral. We have learned that the gauge redundancy does not completely eliminate the path integral over metrics but leaves behind a ﬁnite-dimensional integral over moduli space. It is necessary to reﬁne the discussion from section 3.3 in order to take this into account. The Polyakov path integral for the S-matrix is Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

n [dφ dg] exp(−Sm − λχ) Vdiﬀ×Weyl i=1

d2 σi g(σi )1/2 Vji (ki , σi ) , (5.3.1)

which is the earlier expression (3.5.5) but now written with a general c = ˜c = 26 matter theory, the matter ﬁelds being denoted φ. In gauge-

155

5.3 The measure for moduli

ﬁxing, the integral over metrics and positions is converted to an integral over the gauge group, the moduli, and the unﬁxed positions, [dg] d2n σ → [dζ] dµ t d2n−κ σ .

(5.3.2)

After factoring out the gauge volume, the Jacobian for this transformation becomes the measure on moduli space. The steps are the same as in section 3.3, now taking account of the moduli and the CKVs. In particular, the gauge choice will now ﬁx κ of the vertex operator coordinates, σia → σˆ ia . Label the set of ﬁxed coordinates (a, i) by f. Deﬁne the Faddeev–Popov measure on moduli space by

1 = ∆FP (g, σ)

dµ t diﬀ×Weyl

F

ˆ ζ) [dζ] δ(g − g(t)

δ(σia − σˆ iζa ) .

(5.3.3)

(a,i)∈f

ˆ for at least one By deﬁnition, every metric is (diﬀ×Weyl)-equivalent to g(t) value of t and ζ. The delta function for the ﬁxed coordinates then picks out a unique value of ζ. Actually, as discussed at the end of section 5.1, there may be a residual discrete group of symmetries of ﬁnite order nR , so the delta functions are nonzero at nR points. Inserting the expression (5.3.3) into the path integral (5.3.1) and following the same steps as before leads to Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

F

µ

ˆ σ) ˆ d t ∆FP (g(t),

ˆ × exp −Sm [φ, g(t)] − λχ

n

[dφ]

dσia

(a,i)∈f

ˆ i )1/2 Vji (ki ; σi ) . (5.3.4) g(σ

i=1

The integral over metrics and vertex operator coordinates is now reduced to an integral over the moduli space F for the metric and over the unﬁxed coordinates, with the measure given by ∆FP . In the vertex operators, κ of the positions are ﬁxed. Now we evaluate the Faddeev–Popov measure. The delta functions are nonzero at nR points that are related by symmetry, so we consider one such point and divide by nR . Expand the deﬁnition (5.3.3) of ∆FP near the point. The general metric variation is equal to a local symmetry variation plus a change in the moduli tk , δgab =

µ

ˆ · δσ)gˆ ab . δtk ∂tk gˆ ab − 2(Pˆ 1 δσ)ab + (2δω − ∇

k=1

The inverse Faddeev–Popov determinant is then ˆ σ) ˆ −1 ∆FP (g,

= nR

dµ δt [dδω dδσ] δ(δgab )

δ(δσ a (σˆ i ))

(a,i)∈f

(5.3.5)

156

5 The string S-matrix

= nR

dµ δt dκ x [dβ dδσ]

ˆ + 2πi × exp 2πi(β , 2Pˆ 1 δσ − δtk ∂k g)

a

xai δσ (σˆ i ) ,

(5.3.6)

(a,i)∈f

the inner product in the second line having been deﬁned in exercise 3.2. We have followed the same steps as in the local discussion in section 3.3, writing the delta functions and functionals as integrals over x and βab and be traceless. then integrating out δω to obtain the constraint that βab As before, invert the integral (5.3.6) by replacing all bosonic variables with Grassmann variables: δσ a βab xai δtk

→ → → →

ca , bab , ηai , ξk .

(5.3.7a) (5.3.7b) (5.3.7c) (5.3.7d)

Then with a convenient normalization of the ﬁelds, ˆ σ) ˆ = ∆FP (g,

=

1 nR

1 nR

[db dc] dµ ξ dκ η

1 ˆ + × exp − (b, 2Pˆ 1 c − ξ k ∂k g) ηai ca (σˆ i ) 4π (a,i)∈f

[db dc] exp(−Sg )

µ 1

4π k=1

ˆ (b, ∂k g)

ca (σˆ i ) .

(5.3.8)

(a,i)∈f

In the ﬁnal line we have integrated over the Grassmann parameters ηai and ξ k . The appropriate measure for integration on moduli space is generated by this ghost path integral with insertions. We will not try to keep track of the overall sign at intermediate steps; since this is supposed to be a Jacobian, we implicitly choose the overall sign to give a positive result. The full expression for the S-matrix is then dµ t

Sj1 ...jn (k1 , . . . , kn ) = ×

(a,i)∈f

compact topologies

dσia

F

µ 1

4π k=1

nR

ˆ (b, ∂k g)

[dφ db dc] exp(−Sm − Sg − λχ)

ca (σˆ i )

(a,i)∈f

n

ˆ i )1/2 Vji (ki , σi ) . g(σ

i=1

(5.3.9) This result extends readily to all bosonic string theories — closed and open, oriented and unoriented — the only diﬀerence being which topologies are included and which vertex operators allowed. Eq. (5.3.9) is a useful and

5.3 The measure for moduli

157

elegant result. The complications caused by the moduli and the CKVs are taken into account by the c and b insertions in the path integral. For each ﬁxed coordinate, dσia is replaced by cai , while each metric modulus gives rise to a b insertion. Expression in terms of determinants In eq. (5.3.8) we have represented the Faddeev–Popov determinant as an integral over Grassmann parameters and ﬁelds. We now reduce it to a product of ﬁnite-dimensional and functional determinants. When we go on to study the string S-matrix in coming chapters, this direct path integral approach is just one of the methods we will use. Expand the ghost ﬁelds in suitable complete sets, ca (σ) =

cJ CaJ (σ) ,

bab (σ) =

J

bK BKab (σ) .

(5.3.10)

K

The complete sets CaJ and BKab are deﬁned as follows. The derivative P1 appearing in the ghost action takes vectors into traceless symmetric tensors. Its transpose does the reverse. The ghost action can be written with either, 1 1 T (5.3.11) (b, P1 c) = (P b, c) . 2π 2π 1 We cannot diagonalize P1 in a diﬀ-invariant way because it turns one kind of ﬁeld into another, but we can diagonalize P1T P1 and P1 P1T : Sg =

P1T P1 CaJ = νJ2 CaJ ,

2 P1 P1T BKab = νK BKab .

(5.3.12)

The eigenfunctions can be chosen real, and are normalized in the respective inner products

d2 σ g 1/2 CaJ CJ a = δJJ ,

(5.3.13a)

d2 σ g 1/2 BKab Bab K = δKK .

(5.3.13b)

(P1 P1T )P1 CJ = P1 (P1T P1 )CJ = νJ2 P1 CJ ,

(5.3.14)

(CJ , CJ ) = (BK , BK ) =

Now note that so P1 CJ is an eigenfunction of P1 P1T . In the same way, P1T BK is an eigenfunction of P1T P1 . Thus there is a one-to-one correspondence between the eigenfunctions, except when P1 CJ = 0 or P1T BK = 0. The latter correspond to zero eigenvalues of P1T P1 or P1 P1T . These are just the CKVs and holomorphic quadratic diﬀerentials, and so their numbers are respectively κ and µ. Let us label the eigenfunctions with zero eigenvalue as C0j or B0k , while the nonzero eigenvalues are labeled with J, K = 1, . . ..

158

5 The string S-matrix

For the latter, the normalized eigenfunctions are related BJab =

1 (P1 CJ )ab , νJ

νJ = νJ = 0 .

(5.3.15)

In terms of modes, the ghost path integral ∆FP becomes µ

db0k

κ

dc0j

j=1

k=1

dbJ dcJ exp −

J

µ 1

νJ bJ cJ 2π

4π k =1

ˆ (b, ∂k g)

ca (σi ) .

(a,i)∈f

(5.3.16) From section A.2, a Grassmann integral vanishes unless the variable appears in the integrand. The c0j and b0k do not appear in the action but only in the insertions. In fact, the number of insertions of each type in the integral (5.3.16), κ and µ respectively, just matches the number of ghost zero modes. We have just enough insertions to give a nonzero answer, and only the zero-mode part of the ghost ﬁelds contributes in the insertions. Thus, ∆FP =

µ

db0k

µ b0k

µ k =1

k=1

×

κ

dc0j

j=1

×

J

k =1

ˆ (B0k , ∂k g)

4π

(a,i)∈f

κ

c0j Ca0j (σi )

j =1

νJ bJ cJ dbJ dcJ exp − 2π

.

(5.3.17)

Summing over all ways of saturating the zero-mode integrals with the Grassmann variables generates a ﬁnite-dimensional determinant in each case, while the nonzero modes produce an inﬁnite product, a functional determinant. In all,

∆FP

ˆ (B0k , ∂k g) PTP = det det Ca0j (σi ) det 1 2 1 4π 4π

1/2

.

(5.3.18)

Note that Ca0j (σi ) is a square matrix, in that (a, i) ∈ f runs over κ values as does j. The prime on the functional determinant denotes omission of zero eigenvalues. The Riemann–Roch theorem We can now give a path integral derivation of the Riemann–Roch theorem. For the ghost current of a holomorphic bc system (without ˜b, ˜c), the current conservation anomaly derived in exercise 3.6 is ∇a j a =

1 − 2λ R. 4

(5.3.19)

159

5.4 More about the measure

Noether’s theorem relates conserved currents to invariances. For a current that is not conserved, one ﬁnds by the same argument that δ ([dφ] exp(−S )) i = [dφ] exp(−S ) 2π

d2 σ g 1/2 ∇a j a → −i

2λ − 1 χ. 2

(5.3.20)

The ghost number symmetry acts as δb = −ib, δc = ic. When the path integral is nonvanishing, the transformations of the measure, action, and insertions must cancel. Thus we learn that from the anomaly the number of c minus the number of b insertions is 3χ/2. The path integral calculation relates the number of insertions to the number of zero modes and so gives for the same diﬀerence 1 1 (κ − µ) = (dim ker P1 − dim ker P1T ) . 2 2

(5.3.21)

The factor of 12 appears because we have taken the holomorphic theory only; the antiholomorphic theory gives an equal contribution. Equating the result from the anomaly with the result from the zero modes gives the Riemann–Roch theorem κ − µ = 3χ. For the general bc system one ﬁnds in the same way that dim ker Pn − dim ker PnT = (2n + 1)χ ,

(5.3.22)

the operators being deﬁned in exercise 3.2. 5.4

More about the measure

Here we collect some general properties of the gauge-ﬁxed string amplitude. These are primarily useful for the more formal considerations in chapter 9. Gauge invariance The Faddeev–Popov procedure guarantees that the gauge-ﬁxed amplitude is BRST-invariant and independent of gauge choice, but it is useful to check explicitly that it has all the expected invariance properties. First, it is independent of the choice of coordinates t on moduli space. For new coordinates tk (t), ∂t µ d t = det d t , ∂t µ µ

µ 1 k=1

4π

ˆ = det (b, ∂k g)

∂t ∂t

1 ˆ , (b, ∂k g) 4π k=1

(5.4.1)

160

5 The string S-matrix

and the two Jacobians cancel (up to a sign, which as we have noted must be ﬁxed by hand to give a positive measure). In other words, due to the b-ghost insertions the integrand transforms as a density on moduli space. Second, let us see that the measure is invariant under a Weyl transformation of the gauge slice. Under a Weyl transformation, δ gˆ ab (t; σ) = 2δω(t; σ)gˆ ab (t; σ) ,

(5.4.2)

the variation of the action and measure gives an insertion of the local operator T aa , which vanishes in D = 26 by the equations of motion. We need therefore worry only about the eﬀect on the various insertions. The vertex operator insertions are Weyl-invariant by construction. The insertions ca (σi ) are Weyl-invariant as discussed below eq. (3.3.24). For the b insertions,

(b, ∂k gˆ ) = =

d2 σ gˆ 1/2 bab gˆ ac gˆ bd ∂k gˆ cd

d2 σ gˆ 1/2 bab (gˆ ac gˆ bd ∂k gˆ cd + 2gˆ ab ∂k ω)

ˆ , = (b, ∂k g)

(5.4.3)

the last equality following from tracelessness of b. Third, let us check invariance under an inﬁnitesimal diﬀ transformation, δσ = ξ(t; σ). The extension to a general diﬀ transformation is straightforward. The only terms in the amplitude (5.3.9) that are not immediately invariant are the b insertions and the vertex operators whose coordinates have been ﬁxed. The former transform as ˆ = −2(b, P1 ∂k ξ) δ(b, ∂k g) = −2(P1T b, ∂k ξ) =0

(5.4.4)

using the b equation of motion. The b equation of motion comes from δS /δc = 0, so there will be source terms at the c insertions; these are precisely what is needed to account for the eﬀect of the coordinate transformation on the ﬁxed vertex operators. BRST invariance We now verify the full BRST invariance of the amplitude (5.3.9). The path integral measure and action are known to be invariant from the local analysis, so we have to consider the eﬀect of the BRST transformation on the insertions. In the gauge-ﬁxed path integral (5.3.9), the unintegrated vertex operators are accompanied by a factor of c or ˜cc. This is precisely what we described at the end of section 4.4 for the BRST-invariant vertex operator corresponding to an OCQ state. The integrated vertex operators,

5.4 More about the measure

161

on the other hand, are not BRST-invariant, but we leave it as an exercise to show that they have the BRST variation δB Vm = i∂a (ca Vm ) ,

(5.4.5)

which vanishes upon integration. Finally there is the variation of the b insertion, ˆ = i(T , ∂k g) ˆ . δB (b, ∂k g)

(5.4.6)

This insertion of the energy-momentum tensor just produces a derivative with respect to the modulus tk , so the BRST variation vanishes up to a possible term from the boundary of moduli space. We will examine the boundaries of moduli space in the coming chapters. In most cases there is no surface contribution, due to what is known as the canceled propagator argument. In certain circumstances this argument does not apply, and we will see how to deal with this. It is important to note that the gauge-ﬁxed result (5.3.9) could be written down directly from the requirement of BRST invariance, without reference to the gauge-invariant form. As we saw when evaluating the ghost path integral, there must be at least µ b insertions and κ c insertions, or else the path integral vanishes. In the amplitude (5.3.9) there are precisely the right number of ghost insertions to give a nonzero result. Once we include the necessary b factors, the BRST variation brings in the energymomentum tensor convoluted with the tk -derivative of the metric. This is proportional to a derivative with respect to the moduli, so we get an invariant result only by integrating over moduli space as we have done. The result (5.3.9) can be generalized in various ways, for example to the case that part of the conformal Killing invariance is left unﬁxed. We will illustrate this explicitly in chapter 7 for the example of the torus with no vertex operators. BRST invariance implies that amplitudes for BRST-equivalent states are equal. If we add to any state a null piece QB |χ , the eﬀect is to insert in the path integral the variation δB Vχ ; this integrates to zero. This is important because the physical Hilbert space is identiﬁed with the cohomology, so equivalent states should have equal amplitudes. Measure for Riemann surfaces We derived the Faddeev–Popov measure in the framework where the gauge-ﬁxed world-sheet is described by a moduli-dependent metric. We now recast the result in the Riemann surface framework, where the structure is encoded in moduli-dependent transition functions. To express the measure at a given point t0 in moduli space, let us take a set of coordinate patches, with complex coordinate zm in the mth patch

162

5 The string S-matrix

ˆ 0 ) is Weyland holomorphic transition functions, where the metric g(t equivalent to dzm d¯zm in each patch. Consider now a change in the moduli. We will ﬁrst describe this as a change in the metric with ﬁxed transition functions, and then convert this to a change in the transition functions with ﬁxed metric. In the ﬁrst description, deﬁne the Beltrami diﬀerential 1 µkab = gˆ bc ∂k gˆ ac . 2

(5.4.7)

The b insertion for δtk becomes 1 1 d2 z bzz µk¯zz + b¯z¯z µkz¯z . (b, µk ) = 2π 2π

(5.4.8)

In the second description, after a change δtk in the moduli there will be new coordinates in each patch zm (zm , ¯zm ) . zm = zm + δtk vkm

(5.4.9)

a is a vector index; note that v a is deﬁned only in The superscript on vkm km th the m patch. By the deﬁnition of a Riemann surface, dzm d¯zm is Weylequivalent to the metric at t0 + δt,

dzm d¯zm ∝ dzm d¯zm + δtk µkzm¯zm dzm dzm + µk¯zmzm d¯zm d¯zm

.

(5.4.10)

a is thus related to the Beltrami diﬀerential by The coordinate change vkm ¯zm , µkzm¯zm = ∂zm vkm

zm µk¯zmzm = ∂¯zm vkm .

(5.4.11)

zm is This is the inﬁnitesimal version of Beltrami’s equation. It implies that vkm not holomorphic — otherwise, it would not correspond to a change in the zm has a holomorphic part not determined by Beltrami’s moduli. Also, vkm ¯zm equation, and vkm an antiholomorphic part; these just correspond to the freedom to make holomorphic reparameterizations within each patch. Integrating by parts, the b insertion (5.4.8) becomes

1 1 (b, µk ) = 2π 2πi m

Cm

¯zm zm dzm vkm bzm zm − d¯zm vkm b¯zm ¯zm

,

(5.4.12)

the contour Cm circling the mth patch counterclockwise. By eq. (5.4.9), the derivative of the coordinate with respect to the moduli at a given point is dzm zm = vkm . (5.4.13) dtk The change in the transition functions under a change in moduli is therefore ∂zm ∂zm zn zm zm zm = v − v = vkm − vkn . (5.4.14) km ∂tk zn ∂zn t kn

163

5.4 More about the measure

z

z' 0 C

Fig. 5.3. Coordinate patches z (diagonal hatching) and z (horizontal hatching) with vertex operator at z = 0. In the annular region, z = z + zv .

The contour integrals (5.4.12) around adjacent patches then combine to give 1 1 (b, µk ) = 2π 2πi (mn)

Cmn

∂zm ∂¯zm dzm k bzm zm − d¯zm k b¯zm ¯zm ∂t zn ∂t zn

,

(5.4.15)

which is proportional to the derivatives of the transition functions. The sum runs over all pairs of overlapping patches. The contour Cmn runs between patches m and n, counterclockwise from the point of view of m. The (mn) term is symmetric in m and n, though this is not manifest. Each contour either closes or ends in a triple overlap of patches m, n, and p, where the contours Cmn , Cnp , and Cpm meet at a point. The b insertion is now expressed entirely in terms of the data deﬁning the Riemann surface, the transition functions modulo holomorphic equivalence. As an illustration, we can use this to put the moduli for the metric and for the vertex operator positions on a more equal footing. Consider a vertex operator at position zv in a coordinate system z. Let us make a small new coordinate patch z centered on the vertex operator, as shown in ﬁgure 5.3. We keep the vertex operator at coordinate point z = 0, and encode its position in the transition functions z = z + zv

(5.4.16)

for the annular overlap region. The measure for the moduli zv , ¯zv is given

164

5 The string S-matrix

by the form (5.4.15), where

∂z = −1 . ∂zv z

(5.4.17)

Thus the ghost insertion is C

dz bz z 2πi

C

d¯zm b¯z ¯z = b−1 ˜b−1 · , −2πi

(5.4.18)

where C is any contour encircling the vertex operator as shown in the ﬁgure. The full expression for the S-matrix is then compactly written as S (1; . . . ; n) =

compact topologies

−λχ

e

F

dm t nR

m

Bk

k=1

n

ˆi V

.

(5.4.19)

i=1

ˆ Here Bk is the abbreviation for the b ghost insertion (5.4.15), and V a denotes ˜ccVm for a closed string or ta c Vm for an open string. That is, we are now treating all vertex operators as ﬁxed, having traded their coordinates for extra parameters in the transition functions. The number m of moduli is m = µ + 2nc + no − κ = −3χ + 2nc + no ,

(5.4.20)

where nc and no are the number of closed and open string vertex operators respectively. The expression (5.4.19) shows that the hatted vertex operator, with ˜cc or ta ca , is the basic one. This is what comes out of the state–operator correspondence, and it is BRST-invariant. If the vertex operator is integrated, the ghost insertions (5.4.18) simply remove the ˜cc or ta ca . For example, b−1 ˜b−1 · ˜ccVm = Vm ,

(5.4.21)

leaving the integrated form of the vertex operator. We derived the result (5.4.19) from the Polyakov path integral, so the vertex operators come out in OCQ form, but we may now use arbitrary BRST-invariant vertex operators. Exercises 5.1 Consider the sum (5.1.1), but now over all particle paths beginning and ending at given points in spacetime. In this case ﬁxing the tetrad leaves no coordinate freedom. (a) Derive the analog of the gauge-ﬁxed path integral (5.3.9). (b) Reduce the ghost path integral to determinants as in eq. (5.3.18). (c) Reduce the X µ path integral to determinants (as will be done in section 6.2 for the string).

Exercises

165

(d) Evaluate the ﬁnite and functional determinants and show that the result is the scalar propagator for a particle of mass m. 5.2 Repeat the steps in the previous problem for paths forming a closed loop. In order to ﬁx the residual gauge symmetry, introduce a fake vertex operator by diﬀerentiating with respect to m2 . The answer is given as eq. (7.3.9). If you do both exercises 5.1 and 5.2, discuss the relation between the results for the moduli measure. 5.3 (a) Show from the OPE with the BRST current that the vertex operator ˜ccVm is BRST-invariant for Vm a (1,1) matter tensor. (b) Show that the vertex operator satisﬁes eq. (5.4.5).

6 Tree-level amplitudes

We are now ready to study string interactions. In this chapter we consider the lowest order amplitudes, coming from surfaces with positive Euler number. We ﬁrst describe the relevant Riemann surfaces and calculate the CFT expectation values that will be needed. We next study scattering amplitudes, ﬁrst for open strings and then for closed. Along the way we introduce an important generalization in the open string theory, the Chan–Paton factors. At the end of the section we return to CFT, and discuss some general properties of expectation values. 6.1

Riemann surfaces

There are three Riemann surfaces of positive Euler number: the sphere, the disk, and the projective plane. The sphere The sphere S2 can be covered by two coordinate patches as shown in ﬁgure 6.1. Take the disks |z| < ρ and |u| < ρ for ρ > 1 and join them by identifying points such that u = 1/z .

(6.1.1)

In fact, we may as well take ρ → ∞. The coordinate z is then good everywhere except at the ‘north pole,’ u = 0. We can work mainly in the z patch except to check that things are well behaved at the north pole. We can think of the sphere as a Riemann surface, taking the ﬂat metric in both patches and connecting them by a conformal (coordinate plus Weyl) transformation. Or, we can think of it as a Riemannian manifold, with a globally deﬁned metric. A general conformal gauge metric is ds2 = exp(2ω(z, ¯z ))dzd¯z . 166

(6.1.2)

6.1 Riemann surfaces

167

0 u

z 0 Fig. 6.1. The sphere built from z and u coordinate patches.

Since dzd¯z = |z|4 dud¯u, the condition for the metric to be nonsingular at u = 0 is that exp(2ω(z, ¯z )) fall as |z|−4 for z → ∞. For example, ds2 =

4r2 dzd¯z 4r2 dud¯u = (1 + z¯z )2 (1 + u¯u)2

(6.1.3)

describes a sphere of radius r and curvature R = 2/r2 . According to the general discussion in section 5.2, the sphere has no moduli and six CKVs, so that in particular every metric is (diﬀ×Weyl)equivalent to the round metric (6.1.3). Let us see this at the inﬁnitesimal level. As in eq. (5.2.8) one is looking for holomorphic tensor ﬁelds δgzz (z) and holomorphic vector ﬁelds δz(z). These must be deﬁned on the whole sphere, so we need to consider the transformation to the u-patch, ∂u (6.1.4a) δz = −z −2 δz , ∂z −2 ∂u δguu = δgzz = z 4 δgzz . (6.1.4b) ∂z Any holomorphic quadratic diﬀerential δgzz would have to be holomorphic in z but vanish as z −4 at inﬁnity, and so must vanish identically. A CKV δz, on the other hand, is holomorphic at u = 0 provided it grows no more rapidly than z 2 as z → ∞. The general CKV is then δu =

δz = a0 + a1 z + a2 z 2 , δ¯z = a∗0 + a∗1 ¯z + a∗2 ¯z 2 ,

(6.1.5a) (6.1.5b)

with three complex or six real parameters as expected from the Riemann– Roch theorem. These inﬁnitesimal transformations exponentiate to give the M¨obius

168

6 Tree-level amplitudes

group z =

αz + β γz + δ

(6.1.6)

for complex α, β, γ, δ. Rescaling α, β, γ, δ leaves the transformation unchanged, so we can ﬁx αδ − βγ = 1 and identify under an overall sign reversal of α, β, γ, δ. This deﬁnes the group PSL(2, C). This is the most general coordinate transformation that is holomorphic on all of S2 . It is one-to-one with the point at inﬁnity included. Three of the six parameters correspond to ordinary rotations, forming an S O(3) subgroup of PSL(2, C). The disk It is useful to construct the disk D2 from the sphere by identifying points under a reﬂection. For example, identify points z and z such that z = 1/¯z .

(6.1.7)

In polar coordinates z = reiφ , this inverts the radius and leaves the angle ﬁxed, so the unit disk |z| ≤ 1 is a fundamental region for the identiﬁcation. The points on the unit circle are ﬁxed by the reﬂection, so this becomes a boundary. It is often more convenient to use the conformally equivalent reﬂection z = ¯z .

(6.1.8)

The upper half-plane is now a fundamental region, and the real axis is the boundary. The CKG of the disk is the subgroup of P S L(2, C) that leaves the boundary of the disk ﬁxed. For the reﬂection (6.1.8) this is just the subgroup of (6.1.6) with α, β, γ, δ all real, which is P S L(2, R), the M¨ obius group with real parameters. One CKV is the ordinary rotational symmetry of the disk. Again, all metrics are equivalent — there are no moduli. The projective plane The projective plane RP2 can also be obtained as a Z2 identiﬁcation of the sphere. Identify points z and z with z = −1/¯z .

(6.1.9)

These points are diametrically opposite in the round metric (6.1.3). There are no ﬁxed points and so no boundary in the resulting space, but the space is not oriented. One fundamental region for the identiﬁcation is the unit disk |z| ≤ 1, with points eiφ and −eiφ identiﬁed. Another choice is the upper half-z-plane. There are no moduli. The CKG is the subgroup of

169

6.2 Scalar expectation values

P S L(2, C) that respects the identiﬁcation (6.1.9); this is just the ordinary rotation group S O(3). Both the disk and projective plane have been represented as the sphere with points identiﬁed under a Z2 transformation, or involution. In fact, every world-sheet can be obtained from a closed oriented world-sheet by identifying under one or two Z2 s. The method of images can then be used to obtain the Green’s functions. 6.2

Scalar expectation values

The basic quantities that we need are the expectation values of products of vertex operators. In order to develop a number of useful techniques and points of view we will calculate these in three diﬀerent ways: by direct path integral evaluation, by using holomorphicity properties, and later in the chapter by operator methods. The path integral method has already been used in section 5.3 for the Faddeev–Popov determinant and in the appendix for the harmonic oscillator. Start with a generating functional -

Z[J] =

exp i

.

2

(6.2.1)

d σ J(σ)·X(σ)

for arbitrary Jµ (σ). For now we work on an arbitrary compact twodimensional surface M, and in an arbitrary spacetime dimension d. Expand X µ (σ) in terms of a complete set XI (σ), X µ (σ) =

µ

xI XI (σ) ,

(6.2.2a)

I

∇2 XI = −ωI2 XI ,

M

(6.2.2b)

d2 σg 1/2 XI XI = δII .

(6.2.2c)

Then Z[J] =

dxµI

I,µ

where JIµ =

ωI2 xIµ xIµ exp − + ixµI JIµ 4πα

,

d2 σJ µ (σ)XI (σ) .

(6.2.3)

(6.2.4)

The integrals are Gaussian except for the constant mode

X0 =

2

d σg

1/2

−1/2

,

(6.2.5)

170

6 Tree-level amplitudes

which has vanishing action and so gives a delta function. Carrying out the integrations leaves d d

Z[J] = i(2π) δ (J0 )

I=0

4π 2 α ωI2

d/2

πα JI · JI exp − ωI2

−d/2

−∇2 = i(2π) δ (J0 ) det 4π 2 α 1 × exp − d2 σ d2 σ J(σ) · J(σ )G (σ, σ ) . 2

d d

(6.2.6)

As discussed in section 2.1 and further in section 3.2, the timelike modes x0I give rise to wrong-sign Gaussians and are deﬁned by the contour rotation1 x0I → −ixdI , I = 0. The primed Green’s function excludes the zero mode contribution, G (σ1 , σ2 ) =

2πα I=0

ωI2

XI (σ1 )XI (σ2 ) .

(6.2.7)

It satisﬁes the diﬀerential equation −

1 2 ∇ G (σ1 , σ2 ) = XI (σ1 )XI (σ2 ) 2πα I=0

= g −1/2 δ 2 (σ1 − σ2 ) − X20 ,

(6.2.8)

where the completeness of the XI has been used. The ordinary Green’s function with a delta function source does not exist. It would correspond to the electrostatic potential of a single charge, but on a compact surface the ﬁeld lines from the source have no place to go. The X20 term can be thought of as a neutralizing background charge distribution. The sphere Specializing to the sphere, the solution to the diﬀerential equation (6.2.8) is G (σ1 , σ2 ) = −

1

α ln |z12 |2 + f(z1 , ¯z1 ) + f(z2 , ¯z2 ) , 2

(6.2.9)

A curious factor of i has been inserted into eq. (6.2.6) by hand because it is needed in the S-matrix, but it can be understood formally as arising from the same rotation. If we rotated the entire ﬁeld X 0 → −iX d , the Jacobian should be 1, by the usual argument about rescaling ﬁelds (footnote 1 of the appendix). However, we do not want to rotate x00 , because this mode produces the energy delta function. So we have to rotate it back, giving a factor of i.

171

6.2 Scalar expectation values where

α X20 d2 z exp[2ω(z , ¯z )] ln |z − z |2 + k . (6.2.10) 4 The constant k is determined by the property that G is orthogonal to X0 , but in any case we will see that the function f drops out of all expectation values. It comes from the background charge, but the delta function from the zero-mode integration forces overall neutrality, J0µ = 0, and the background makes no net contribution. Now consider the path integral with a product of tachyon vertex operators, f(z, ¯z ) =

AnS2 (k, σ) =

eik1 ·X(σ1 )

r

eik2 ·X(σ2 ) . . . eikn ·X(σn )

r

r

S2

(6.2.11)

.

This corresponds to J(σ) =

n

ki δ 2 (σ − σi ) .

(6.2.12)

i=1

The amplitude (6.2.6) then becomes AnS2 (k, σ) = iCSX2 (2π)d δ d (

i ki )

n

n 1 ki ·kj G (σi , σj ) − ki2 Gr (σi , σi ) × exp − 2 i,j=1 i=1

(6.2.13)

.

i<j

The constant here is

CSX2 = X−d 0

−∇2 det 2 4π α

−d/2

(6.2.14)

. S2

The determinant can be regularized and computed, but we will not need to do this explicitly. We are employing here the simple renormalization used in section 3.6, so that the self-contractions involve Gr (σ, σ ) = G (σ, σ ) +

α ln d2 (σ, σ ) . 2

(6.2.15)

Note that Gr (σ, σ) = 2f(z, ¯z ) + α ω(z, ¯z )

(6.2.16)

is ﬁnite by design. The path integral on the sphere is then

α 2 AnS2 (k, σ) = iCSX2 (2π)d δ d ( i ki ) exp − k ω(σi ) 2 i i

n

|zij |α ki ·kj .

i,j=1 i<j

(6.2.17) The function f has dropped out as promised. The dependence on the conformal factor ω(σ) is precisely that found in section 3.6 from the Weyl

172

6 Tree-level amplitudes

anomaly in the vertex operator. It will cancel the variation of g 1/2 for an on-shell operator.2 We can take a metric that is ﬂat in a large region containing all the vertex operators (‘pushing the curvature to inﬁnity’) and the term in ω(σi ) drops out. Higher vertex operators are exponentials times derivatives of X µ , so we also need n

eiki ·X(zi ,¯zi )

i=1

p r

∂X µj (zj )

j=1

q

¯ νk (¯z ) ∂X k

k=1

(6.2.18)

. S2

¯ This is given by summing over all contractions, where every ∂X or ∂X ¯ must be contracted either with an exponential or with another ∂X or ∂X. The XX contraction is simply − 12 α ln |z|2 , the fs again dropping out in the ﬁnal expression. The result can be summarized as iCSX2 (2π)d δ d ( ×

i ki )

p

n

|zij |α ki ·kj

i,j=1 i<j

v

µj

(zj )

+q

µj

j=1

(zj )

q

νk

˜v

(¯zk )

+ q˜

νk

(¯zk )

(6.2.19)

k=1

Here v µ (z) = −i

n α kiµ , 2 i=1 z − zi

˜v µ (¯z ) = −i

n α kiµ 2 i=1 ¯z − ¯zi

(6.2.20)

come from the contractions with the exponentials. The expectation values of q µ = ∂X µ − v µ are given by the sum over all contractions using −η µν (z − z )−2 α /2, and those of q˜ν by the conjugate. Now we repeat the calculation in a diﬀerent way, using holomorphicity. As an example consider

∂X µ (z1 )∂X ν (z2 ) S2 . The OPE determines this to be α η µν

1 S2 + g(z1 , z2 ) , − 2 2z12

(6.2.21)

(6.2.22)

where g(z1 , z2 ) is holomorphic in both variables. In the u-patch, ∂u X µ = −z 2 ∂z X µ ,

(6.2.23)

so the condition of holomorphicity at u = 0 is that expectation values of ∂z X µ fall as z −2 at inﬁnity. More generally, a tensor of weight (h, 0) must 2

The constant CSX also depends on the conformal factor, because the X µ CFT by itself has 2 nonzero central charge, but in the full string theory this will cancel.

173

6.2 Scalar expectation values

behave as z −2h at inﬁnity. Focusing on the z1 -dependence in eq. (6.2.22) at ﬁxed z2 , this implies that g(z1 , z2 ) falls as z1−2 at inﬁnity and so vanishes by holomorphicity. This determines the expectation value up to normalization. Comparing with the path integral result (6.2.19), there is agreement, though the normalization 1 S2 is seen to diverge as δ(0). This is just the zero-mode divergence from the inﬁnite volume of spacetime. To obtain the expectation value of the product of vertex operators AnS2 (k, σ) =

n

: eiki ·X(zi ,¯zi ) :

i=1

(6.2.24) S2

by this method is less direct, because the exponentials are not holomorphic. In fact they can be factored into holomorphic and antiholomorphic parts, but this is subtle and is best introduced in operator language as we will do in chapter 8. We use the holomorphicity of the translation current. Consider the expectation value with one current ∂X µ (z) added. The OPE of the current with the vertex operators determines the singularities in z, µ

∂X (z)

n

iki ·X(zi ,¯zi )

:e

i=1

=−

: S2

n iα n kiµ AS2 (k, σ) 2 z − zi i=1

+ terms holomorphic in z.

(6.2.25)

Now look at z → ∞. The condition that ∂u X µ be holomorphic at u = 0 again requires that the expectation value vanish as z −2 when z → ∞. The holomorphic term in eq. (6.2.25) must then vanish, and from the vanishing of the order z −1 term we recover momentum conservation, AnS2 (k, σ)

n µ

ki = 0.

(6.2.26)

i=1

Let us also say this in a slightly diﬀerent way. Consider the contour integral of the spacetime translation current 1 p = 2πi

µ

C

(dz jzµ − d¯z j¯zµ ) ,

(6.2.27)

where the contour C encircles all the exponential operators. There are two ways to evaluate this. The ﬁrst is to contract C until it is a small circle around each vertex operator, which picks out the (z − zi )−1 term from each OPE and gives ni=1 kiµ . The second is to expand it until it is a small circle in the u-patch: by holomorphicity at u = 0 it must vanish. This sort of ‘contour-pulling’ argument is commonly used in CFT.

174

6 Tree-level amplitudes

We have used the OPE to determine the singularities and then used holomorphicity to get the full z-dependence. Now we look at the second term of the OPE as z → z1 . Expanding the result (6.2.25) gives

n iα k1µ kiµ − AnS2 (k, σ) + + O(z − z1 ) . 2 z − z1 i=2 z1 − zi

(6.2.28)

Contracting this with ik1µ , the term of order (z − z1 )0 must agree with the OPE ik1 ·∂X(z) : eik1 ·X(z1 ,¯z1 ) : α k12 = : eik1 ·X(z1 ,¯z1 ) : + ∂z1 : eik1 ·X(z1 ,¯z1 ) : + O(z − z1 ) . 2(z − z1 )

(6.2.29)

This implies ∂z1 AnS2 (k, σ) =

n k1 ·ki α n . AS2 (k, σ) 2 z1i i=2

(6.2.30)

Integrating, and using the conjugate equation and momentum conservation, determines the path integral up to normalization, AnS2 (k, σ) ∝ δ d (

i ki )

n

|zij |α ki ·kj ,

(6.2.31)

i,j=1 i<j

in agreement with the ﬁrst method. Note that to compare we must push the curvature to inﬁnity (ω(σi ) = 0) so that [ ]r = : :. It is worth noting that the intermediate steps in the path integral method depend in a detailed way on the particular choice of Riemannian metric. The metric is needed in order to preserve coordinate invariance in these steps. A diﬀerent, Weyl-equivalent, metric gives a diﬀerent Laplacian and diﬀerent eigenfunctions, though at the very end this dependence must drop out in string theory. The second method uses only the basic data of a Riemann surface, its holomorphic structure. The disk The generalization to the disk is straightforward. We will represent the disk by taking the above representation of the sphere and restricting z to the upper half of the complex plane. The Neumann boundary term is accounted for by an image charge, G (σ1 , σ2 ) = −

α α ln |z1 − z2 |2 − ln |z1 − ¯z2 |2 , 2 2

(6.2.32)

175

6.2 Scalar expectation values up to terms that drop out due to momentum conservation. Then n

iki ·X(zi ,¯zi )

:e

= iCDX2 (2π)d δ d (

:

i=1

i ki )

n

2

|zi − ¯zi |α ki /2

i=1

D2

×

n

|zi − zj |α ki ·kj |zi − ¯zj |α ki ·kj .

(6.2.33)

i,j=1 i<j

For expectation values with ∂a X µ s one again sums over contractions, ¯ ν (¯z ) (or now using the Green’s function (6.2.32). Note that ∂X µ (z)∂X µ ν q (¯z )) has a nonzero contraction. q (z)˜ For two points on the boundary, the two terms in the Green’s function (6.2.32) are equal, and the Green’s function diverges at zero separation even after normal ordering subtracts the ﬁrst term. For this reason, boundary operators must be deﬁned with boundary normal ordering, where the subtraction is doubled:

X µ (y1 )X ν (y2 ) = X µ (y1 )X ν (y2 ) + 2α η µν ln |y1 − y2 | ,

(6.2.34)

with y denoting a coordinate on the real axis. The combinatorics are the same as for other forms of normal ordering. Boundary normal-ordered expectation values of boundary operators have the same good properties (nonsingularity) as conformal normal-ordered operators in the interior. An expectation value with exponentials both in the interior and on the boundary, each having the appropriate normal ordering, is given by taking the appropriate limit of the interior result (6.2.33) and dropping the 2 |zi − ¯zi |α ki /2 factor for the boundary operators. Explicitly, for exponentials all on the boundary, n

iki ·X(yi )

e

= iCDX2 (2π)d δ d (

i=1

i ki )

More generally,

eiki ·X(yi )

i=1

p

(6.2.35)

∂y X µj (yj )

j=1

×

|yi − yj |2α ki ·kj .

i,j=1 i<j

D2

n

n

n i,j=1 i<j

|yij |

2α ki ·kj

= iCDX2 (2π)d δ d (

D p2

v

µj

(yj )

+q

j=1

µj

(yj )

i ki )

,

(6.2.36)

D2

where now v µ (y) = −2iα

n i=1

kiµ y − yi

and the qs are contracted using −2α (y − y )−2 η µν .

(6.2.37)

176

6 Tree-level amplitudes The projective plane

The method of images gives the Green’s function as G (σ1 , σ2 ) = − Now

n

α α ln |z1 − z2 |2 − ln |1 + z1 ¯z2 |2 . 2 2

iki ·X(zi ,¯zi )

:e

:

i=1

=

X iCRP (2π)d δ d ( 2

i ki )

n

(6.2.38)

2

|1 + zi ¯zi |α ki /2

i=1

RP2

×

n

|zi − zj |α ki ·kj |1 + zi ¯zj |α ki ·kj ,

(6.2.39)

i,j=1 i<j

and so on for more general expectation values. There is no boundary because the involution has no ﬁxed points — there is no way for a point z to approach its image −¯z −1 . 6.3

The bc CFT The sphere

The path integral for the ghosts has already been set up in section 5.3. According to the Riemann–Roch theorem, the simplest nonvanishing expectation value will be

c(z1 )c(z2 )c(z3 )˜c(¯z4 )˜c(¯z5 )˜c(¯z6 ) S2 .

(6.3.1)

Up to normalization (a functional determinant), the result (5.3.18) was just the zero-mode determinant, det Ca0j (σi ) .

(6.3.2)

The six CKVs were found in section 6.1. In a complex basis they are Cz = 1, z, z 2 , C¯z = 0 , Cz = 0 , C¯z = 1, ¯z , ¯z 2 .

(6.3.3a) (6.3.3b)

In this basis the determinant splits into two 3 × 3 blocks, and the expectation value becomes 1 CSg2 det z1 z2 1

1 1 1 1 1 z2 z3 det ¯z4 ¯z5 ¯z6 = CSg2 z12 z13 z23 ¯z45 ¯z46 ¯z56 . (6.3.4) ¯ z42 ¯z52 ¯z62 z22 z32

The constant CSg2 includes the functional determinant and also a ﬁnitedimensional Jacobian (independent of the positions) that arises because the basis (6.3.3) is not orthonormal.

177

6.3 The bc CFT

This is the only bc path integral that we will actually need for the tree-level amplitudes, but for completeness we give p+3 i=1

c(zi )

p

b(zj )

= CSg2

· (anti)

j=1

S2

zp+1,p+2 zp+1,p+3 zp+2,p+3 · (anti) (z1 − z1 ) . . . (zp − zp ) ± permutations ,

(6.3.5)

obtained by contracting the bs with cs. The antiholomorphic part ‘anti’ has the same form. We are being a little careless with the overall sign; again, for the Faddeev–Popov determinant we will in any case take an absolute value. To give an alternative derivation using holomorphicity, we again consider ﬁrst the conservation law, derived /by inserting into the amplitude (6.3.5) the ghost number contour integral C dz j(z)/2πi, where C encircles all the vertex operators and j = − : bc : is again the ghost number current. From the OPE with b and c, this just counts the number of c minus the number of b ﬁelds, giving nc − nb times the original amplitude. Now pull the contour into the u-patch using the conformal transformation (2.5.17), C

dz jz = − 2πi

C

du ju + 3 → 3 . 2πi

(6.3.6)

The shift +3 comes about because j is not a tensor, having a z −3 term in its OPE with T . In the last step we use holomorphicity in the u-patch. The nonzero amplitudes therefore have nc − nb = 3 and similarly n˜c − n˜b = 3, in agreement with the Riemann–Roch theorem. From the OPE, the ghost expectation value (6.3.1) is holomorphic in each variable, and it must have a zero when two identical anticommuting ﬁelds come together. Thus it must be of the form ˜ z4 , ¯z5 , ¯z6 ) , z12 z13 z23 ¯z45 ¯z46 ¯z56 F(z1 , z2 , z3 )F(¯

(6.3.7)

˜ antiholomorphic functions of the positions. with F holomorphic and F As z1 → ∞ this goes as z12 F. However, c is a tensor of weight −1, so the amplitude can be no larger than z12 at inﬁnity. Thus F(zi ) must be ˜ we independent of z1 , and so also of z2 and z3 ; arguing similarly for F obtain again the result (6.3.4). The same argument gives for the general case (6.3.5) the result CSg2

p+3 i,i =1 i*
*

zii

p j,j =1 j<j

zjj

p p+3

(zi − zj )−1 · (anti) ,

(6.3.8)

i=1 j=1

which has the correct poles, zeros, and behavior at inﬁnity. The permutations (6.3.5) evidently sum up to give this single term.

178

6 Tree-level amplitudes

We have considered the λ=2 theory that is relevant to the ghosts of the bosonic string, but either method generalizes readily to any λ. The disk The simplest way to obtain the bc amplitudes on the disk is with the doubling trick. As in eq. (2.7.30), we can represent both the holomorphic and antiholomorphic ﬁelds in the upper half-plane by holomorphic ﬁelds in the whole plane, using ˜b(¯z ) = b(z ) , ˜c(¯z ) = c(z ) , z = ¯z , Im z > 0 . (6.3.9) The expectation value of the holomorphic ﬁelds follows as on the sphere,

c(z1 )c(z2 )c(z3 ) D2 = CDg 2 z12 z13 z23 for all z. Then for example

c(z1 )c(z2 )˜c(¯z3 ) D2 = c(z1 )c(z2 )c(z3 )

(6.3.10)

D2

= CDg 2 z12 (z1 − ¯z3 )(z2 − ¯z3 ) .

(6.3.11)

More general correlators are obtained in the same way. The projective plane The doubling trick again can be used. The involution z = −¯z −1 implies that ˜b(¯z ) =

∂z ∂¯z

2

b(z ) = z 4 b(z ) ,

˜c(¯z ) =

∂z ∂¯z

−1

c(z ) = z −2 c(z ) . (6.3.12)

Again g

c(z1 )c(z2 )c(z3 ) RP2 = CRP z z z 2 12 13 23

and so

c(z1 )c(z2 )˜c(¯z3 ) RP2 = z3−2 c(z1 )c(z2 )c(z3 ) = 6.4

g CRP z (1 2 12

(6.3.13)

RP2

+ z1 ¯z3 )(1 + z2 ¯z3 ) .

(6.3.14)

The Veneziano amplitude

Open string amplitudes are slightly simpler than closed string amplitudes, so we begin with these. We represent the disk as the upper half-plane so the boundary coordinate y is real. There are no moduli. After ﬁxing the metric, the CKG P S L(2, R) can be used to ﬁx the three vertex operators to arbitrary positions y1 , y2 , y3 on the boundary, except that this group does not change

179

6.4 The Veneziano amplitude

the cyclic ordering of the vertex operators so we must sum over the two orderings. For three open string tachyons on the disk, the general expression (5.3.9) for the string S-matrix thus reduces to SD2 (k1 ; k2 ; k3 ) = go3 e−λ

c1 eik1 ·X (y1 ) c1 eik2 ·X (y2 ) c1 eik3 ·X (y3 )

+ (k2 ↔ k3 ) ,

D2

(6.4.1)

each ﬁxed coordinate integration being replaced by the corresponding c-ghost. Each vertex operator includes a factor of go , the open string coupling. The factor e−λ is from the Euler number term in the action. Of course go and eλ are related, go2 ∝ eλ , but we will determine the constant of proportionality as we go along. The expectation values in (6.4.1) were found in the previous sections (again, we take the absolute value for the ghosts), giving SD2 (k1 ; k2 ; k3 ) = igo3 CD2 (2π)26δ 26 ( ×|y12 |

1+2α k1 ·k2

|y13 |

i ki ) 1+2α k1 ·k3

|y23 |1+2α k2 ·k3 + (k2 ↔ k3 ), (6.4.2)

where CD2 = e−λ CDX2 CDg 2 . Momentum conservation and the mass-shell condition ki2 = 1/α imply that 2α k1 ·k2 = α (k32 − k12 − k22 ) = −1

(6.4.3)

and the same for the other ki · kj , so this reduces to SD2 (k1 ; k2 ; k3 ) = 2igo3 CD2 (2π)26δ 26 (

i ki )

(6.4.4)

.

This is independent of the gauge choice yi , which is of course a general property of the Faddeev–Popov procedure. The Weyl invariance is crucial here — if the vertex operators were taken oﬀ the mass shell, the amplitude would depend on the choice of yi . We could have used independence of the yi to determine the Faddeev– Popov determinant without calculation. Using the mass-shell condition and momentum conservation, the X µ expectation value is proportional to |y12 y13 y23 |−1 , and so the measure must be reciprocal to this. This same measure then applies for n > 3 vertex operators, because in all cases three positions are ﬁxed. The four-tachyon amplitude is obtained in the same way, SD2 (k1 ; k2 ; k3 ; k4 ) = go4 e−λ =

∞

−∞

dy4

3

igo4 CD2 (2π)26δ 26 (

c1 (yi )eiki ·X(yi ) eik4 ·X(y4 )

i=1 i ki )|y12 y13 y23 |

∞ −∞

dy4

+ (k2 ↔ k3 )

|yij |2α ki ·kj

i<j

+ (k2 ↔ k3 ) .

(6.4.5)

180

6 Tree-level amplitudes

3

2

3

2

3

4

(a) 4

1

(b) 1

4

(c) 1

2

2

3

2

3

2

4

(d) 4

1

(e) 1

4

(f) 1

3

Fig. 6.2. The six cyclically inequivalent orderings of four open string vertex operators on the disk. The coordinate y increases in the direction of the arrow, except that at point 3 it jumps from +∞ to −∞.

This is again independent of y1,2,3 after a change of variables (a M¨ obius transformation) on y4 . It is convenient to take y1 = 0, y2 = 1, and y3 → ∞. The amplitude is conventionally written in terms of the Mandelstam variables s = −(k1 + k2 )2 ,

t = −(k1 + k3 )2 ,

u = −(k1 + k4 )2 .

(6.4.6)

These are not independent: momentum conservation and the mass-shell condition imply that 4 m2i = − . (6.4.7) s+t+u= α i Using 2α ki · kj = −2 + α (ki + kj )2 , the amplitude becomes SD2 (k1 ; k2 ; k3 ; k4 ) = igo4 CD2 (2π)26δ 26 ( ∞

×

−∞

dy4 |y4 |

−α u−2

i ki )

|1 − y4 |−α t−2 + (t → s) .

(6.4.8)

The integral splits into three ranges, −∞ < y4 < 0, 0 < y4 < 1, and 1 < y4 < ∞. For these three ranges the vertex operators are ordered as in ﬁgures 6.2(a), (b), and (c) respectively. M¨ obius invariance can be used to take each of these ranges into any other, so they give contributions that are equal up to interchange of vertex operators. The (t ↔ s) term gives ﬁgures 6.2(d), (e), and (f). In all, SD2 (k1 ; k2 ; k3 ; k4 ) = 2igo4 CD2 (2π)26δ 26 (

i ki )

I(s, t) + I(t, u) + I(u, s) , (6.4.9)

181

6.4 The Veneziano amplitude

(a)

3

4

2

1

(b)

3

4

1

2

3

4

1

2

(c)

Fig. 6.3. Processes giving poles in the (a) s-, (b) t-, and (c) u-channels.

where

1

I(s, t) = 0

dy y −α s−2 (1 − y)−α t−2 .

(6.4.10)

The three terms come from ﬁgures 6.2(c), (f), 6.2(b), (d), and 6.2(a), (e) respectively. The integral I(s, t) converges if α s < −1 and α t < −1. As α s → −1, the integral diverges at y = 0. To study the divergence, take a neighborhood of y = 0 and approximate the integrand: r

I(s, t) = 0

dy y −α s−2 + terms analytic at α s = −1

r−α s−1 =− + terms analytic at α s = −1 αs+1 1 =− (6.4.11) + terms analytic at α s = −1 . αs+1 In (6.4.11), we have evaluated the integral in the convergent region. We see that the divergence is a pole at s = −1/α , the mass-squared of the open string tachyon. The variable s is just the square of the center-ofmass energy for scattering 1 + 2 → 3 + 4, so this pole is a resonance due to an intermediate tachyon state. Again, it is an artifact of the bosonic string that this lightest string state is tachyonic, and not relevant to the discussion. The pole is due to the process shown in ﬁgure 6.3(a), in which tachyons 1 and 2 join to become a single tachyon, which then splits into tachyons 3 and 4. Because the singularity at α s = −1 is just a pole, I(s, t) can be analytically continued past this point into the region α s > −1. The amplitude is deﬁned via this analytic continuation. The divergence of the amplitude at the pole is an essential physical feature of the amplitude, a resonance corresponding to propagation of the intermediate string state over long spacetime distances. The divergence of the integral past the pole is not; it is just an artifact of this particular integral representation of the amplitude. The continuation poses no problem. In fact, we will see that every

182

6 Tree-level amplitudes

string divergence is of this same basic form, so this one kind of analytic continuation removes all divergences — except of course for the poles themselves. The pole is on the real axis, so we need to deﬁne it more precisely. The correct prescription for a Minkowski process is 1 1 ≡ +1 α s + 1 + i 1 − iπδ(α s + 1) , (6.4.12) ≡P αs+1 where P denotes the principal value. Unitarity (which we will develop more systematically in chapter 9) requires this pole to be present and determines its coeﬃcient in terms of the amplitude for two tachyons to scatter into one: d26 k SD2 (k1 ; k2 ; k)SD2 (−k; k3 ; k4 ) SD2 (k1 ; k2 ; k3 ; k4 ) = i (2π)26 −k 2 + α−1 + i α s

+ terms analytic at k 2 = 1/α . (6.4.13) Gathering together the factors in the four-tachyon amplitude, including an equal contribution to the pole from I(u, s), and using the three-tachyon result (6.4.4), the condition (6.4.13) gives CD2 = e−λ CDX2 CDg 2 =

1 α go2

(6.4.14)

.

The three-tachyon amplitude is then 2igo (2π)26δ 26 ( i ki ) . (6.4.15) α The various functional determinants have dropped out. Using unitarity, all normalizations can be expressed in terms of the constant go appearing in the vertex operators. The determinants can in fact be computed by careful regularization and renormalization, and the relative normalizations of diﬀerent topologies agree with those from unitarity. Continuing past the pole, we encounter further singularities. Taylor expanding the integrand at y = 0 gives SD2 (k1 ; k2 ; k3 ) =

r

I(s, t) =

dy y −α s−2 + (α t + 2)y −α s−1 + . . .

.

(6.4.16)

0

The second term gives a pole at α s = 0, u−t I(s, t) = (6.4.17) + terms analytic at α s = 0 . 2s From the further terms in the Taylor expansion, the amplitude has poles at α s = −1, 0, 1, 2, . . . .

(6.4.18)

183

6.4 The Veneziano amplitude

These are precisely the positions of the open string states. The integral I(s, t) also has poles in the variable t at the same positions (6.4.18), coming from the endpoint y = 1. These are due to the process of ﬁgure 6.3(b). The other two terms in the amplitude (6.4.9) give further contributions to the s- and t-channel poles, and also give poles in the u-channel, ﬁgure 6.3(c). Because the residue at s = 0 in (6.4.17) is odd in u − t, this pole actually cancels that from I(s, u), as do all poles at even multiples of 1/α . This will not be true for the more general open string theories to be introduced in the next section. Deﬁne the Euler beta function 1

B(a, b) = 0

dy y a−1 (1 − y)b−1 ,

(6.4.19)

so that αo (x) = 1 + α x .

I(s, t) = B(−αo (s), −αo (t)) ,

(6.4.20)

This can be expressed in terms of gamma functions. Deﬁning y = v/w for ﬁxed w gives w

a+b−1

w

B(a, b) = 0

dv v a−1 (w − v)b−1 .

Multiplying both sides by e−w , integrating Γ(a + b)B(a, b) =

∞

dv v a−1 e−v

0

∞ 0

∞ 0

(6.4.21)

dw, and regrouping gives

d(w − v)(w − v)b−1 e−(w−v)

= Γ(a)Γ(b) .

(6.4.22)

The four-tachyon amplitude is then 2igo2 (2π)26δ 26 ( i ki ) α

× B(−αo (s), −αo (t)) + B(−αo (s), −αo (u)) + B(−αo (t), −αo (u)) ,

SD2 (k1 ; k2 ; k3 ; k4 ) =

(6.4.23) where Γ(−α x − 1)Γ(−α y − 1) . (6.4.24) Γ(−α x − α y − 2) This is the Veneziano amplitude, originally written to model certain features of strong interaction phenomenology. The high energy behavior of the Veneziano amplitude is important. There are two regions of interest, the Regge limit, B(−αo (x), −αo (y)) =

s→∞,

t ﬁxed ,

(6.4.25)

t/s ﬁxed .

(6.4.26)

and the hard scattering limit, s→∞,

184

6 Tree-level amplitudes

If we consider the scattering process 1 + 2 → 3 + 4 (so that k10 and k20 are positive and k30 and k40 negative), then in the 1-2 center-of-mass frame, θ θ , u = (4m2 − E 2 ) cos2 , (6.4.27) 2 2 where E is the center-of-mass energy and θ is the angle between particle 1 and particle 3. The Regge limit is high energy and small angle, while the hard scattering limit is high energy and ﬁxed angle. Using Stirling’s approximation, Γ(x+1) ≈ xx e−x (2π x)1/2 , the behavior in the Regge region is s = E2 ,

t = (4m2 − E 2 ) sin2

SD2 (k1 ; k2 ; k3 ; k4 ) ∝ sαo (t) Γ(−αo (t)) ,

(6.4.28)

where again αo (t) = α t + 1. That is, the amplitude varies as a power of s, the power being t-dependent. This is Regge behavior. At the poles of the gamma function, the amplitude is an integer power of s, corresponding to exchange of a string of integer spin αo (t). In the hard scattering limit, %

&

SD2 (k1 ; k2 ; k3 ; k4 ) ≈ exp −α (s ln sα + t ln tα + u ln uα ) = exp[−α sf(θ)] , (6.4.29) where θ θ θ θ f(θ) ≈ − sin2 ln sin2 − cos2 ln cos2 (6.4.30) 2 2 2 2 is positive. The result (6.4.29) is notable. High energy, ﬁxed angle scattering probes the internal structure of the objects being scattered. Rutherford discovered the atomic nucleus with hard alpha–atom scattering. Hard electron–nucleon scattering at SLAC revealed the quark constituents of the nucleon. In quantum ﬁeld theory, hard scattering amplitudes fall as a power of s. Even a composite object like the nucleon, if its constituents are pointlike, has power law amplitudes. The exponential falloﬀ (6.4.29) is very much softer. The result (6.4.29) suggests a smooth object of size α 1/2 , as one would expect. We started with the three-particle amplitude, skipping over the zero-, one-, and two-particle amplitudes. We will discuss these amplitudes, and their interpretation, in section 6.6. 6.5

Chan–Paton factors and gauge interactions

In this section we will consider the interactions of the massless vector state of the open string. To make the discussion a bit more interesting, we ﬁrst introduce a generalization of the open string theory. At the end of chapter 3 we introduced a very large class of bosonic string theories, but in this ﬁrst look at the interactions we are focusing

6.5 Chan–Paton factors and gauge interactions

185

j

i

Fig. 6.4. Open string with Chan–Paton degrees of freedom.

on the simplest case of 26 ﬂat dimensions. One can think of this in terms of symmetry: this theory has the maximal 26-dimensional Poincar´e invariance. In the closed bosonic string this is the unique theory with this symmetry. An outline of the proof is as follows. The world-sheet Noether currents for spacetime translations have components of weights (1, 0) and (0, 1). By an argument given in section 2.9, these currents are then holomorphic in z or ¯z . We have seen in the calculations in this chapter that this is enough to determine all the expectation values. In open string theory, however, there is a generalization. The open string has boundaries, endpoints. In quantum systems with distinguished points it is natural to have degrees of freedom residing at those points in addition to the ﬁelds propagating in the bulk. At each end of the open string let us add a new degree of freedom, known as a Chan–Paton degree of freedom, which can be in one of n states. A basis of string states is then |N; k; ij ,

(6.5.1)

where i and j denote the states of the left- and right-hand endpoints, running from 1 to n. The energy-momentum tensor is deﬁned to be the same as before, with no dependence on the new degrees of freedom. Conformal invariance is therefore automatic. Poincar´e invariance is automatic as well, as the Chan–Paton degrees of freedom are invariant. Although these new degrees of freedom have trivial world-sheet dynamics, they will have a profound eﬀect on the spacetime physics. In string theories of the strong interaction, the motivation for this was to introduce S U(3) ﬂavor quantum numbers: the endpoints are like quarks and antiquarks, connected by a color-electric ﬂux tube. Now we are motivating it in the general framework of considering all possibilities with given symmetries. We will give a new interpretation to the Chan– Paton degrees of freedom in chapter 8, and a possible further reﬁnement in chapter 14. There are now n2 scalar tachyons, n2 massless vector bosons, and so forth. The n2 Hermitian matrices λaij , normalized to Tr (λa λb ) = δ ab ,

(6.5.2)

are a complete set of states for the two endpoints. These are the representation matrices of U(n), so one might guess that the massless vectors are

186

6 Tree-level amplitudes

associated with a U(n) gauge symmetry; we will soon see that this is the case. Deﬁne the basis |N; k; a =

n

|N; k; ij λaij .

(6.5.3)

i,j=1

Now consider the four-tachyon amplitude shown in ﬁgure 6.2(a), in which the vertex operators are arranged in the cyclic order 1234. Because the Chan–Paton degrees of freedom do not appear in the Hamiltonian, their state does not evolve between the vertex operators: the right-hand endpoint of tachyon 1 must be in the same state as the left-hand endpoint of tachyon 2, and so forth. Thus, the amplitude 6.2(a) will now contain an additional factor of Tr (λa1 λa2 λa3 λa4 )

(6.5.4)

from the overlap of the Chan–Paton wavefunctions for each tachyon. This rule generalizes to an arbitrary amplitude: each vertex operator now contains a Chan–Paton factor λaij from the wavefunction of the endpoint degrees of freedom, and the amplitude for each world-sheet is multiplied by a trace of the Chan–Paton factors around each boundary. The three-tachyon amplitude becomes SD2 (k1 , a1 ; k2 , a2 ; k3 , a3 ) = igo (2π)26δ 26 ( i ki )Tr (λa1 λa2 λa3 + λa1 λa3 λa2 ) , (6.5.5) α the two cyclic orderings now having diﬀerent Chan–Paton traces. The four-tachyon amplitude is igo2 (2π)26δ 26 ( i ki ) α

× Tr (λa1 λa2 λa4 λa3 + λa1 λa3 λa4 λa2 )B(−αo (s), −αo (t))

SD2 (k1 , a1 ; k2 , a2 ; k3 , a3 ; k4 , a4 ) =

+ Tr (λa1 λa3 λa2 λa4 + λa1 λa4 λa2 λa3 )B(−αo (t), −αo (u))

+ Tr (λa1 λa2 λa3 λa4 + λa1 λa4 λa3 λa2 )B(−αo (s), −αo (u)) .

(6.5.6)

Considering again the unitarity relation (6.4.13), the pole at s = −1/α acquires a factor of 1 a1 a2 a3 a4 Tr {λ , λ }{λ , λ } 4 on the left and a factor of 1 a1 a2 a a3 a4 a Tr {λ , λ }λ Tr {λ , λ }λ , 4 a

(6.5.7)

(6.5.8)

187

6.5 Chan–Paton factors and gauge interactions

on the right, the sum being over the Chan–Paton wavefunction of the intermediate state. The completeness of the λaij and the normalization (6.5.2) imply that Tr (Aλa ) Tr (Bλa ) = Tr (AB)

(6.5.9)

for any matrices A and B, and so the amplitudes are still unitary. Gauge interactions The amplitude for a gauge boson and two tachyons is SD2 (k1 , a1 , e1 ; k2 , a2 ; k3 , a3 ) = −igo go2 e−λ e1µ ˙ µ eik1 ·X (y1 ) c1 eik2 ·X (y2 ) c1 eik3 ·X (y3 ) × c1 X

D2

Tr (λa1 λa2 λa3 )

+ (k2 , a2 ) ↔ (k3 , a3 ) . (6.5.10) We have used the gauge boson vertex operator (3.6.26), but for now are allowing an independent normalization constant go . Using the results of section 6.2, the X path integral is

˙ µ eik1 ·X (y1 ) eik2 ·X (y2 ) eik3 ·X (y3 ) X ×iCDX2 (2π)26δ 26 (

i ki )|y12 |

2α k1 ·k2

D2

= −2iα

µ

k3µ k2 + y12 y13

|y13 |2α k1 ·k3 |y23 |2α k2 ·k3 .

(6.5.11)

Using momentum conservation, the mass-shell conditions, and the physical state condition k1 ·e1 = 0, the amplitude becomes SD2 (k1 , a1 , e1 ; k2 , a2 ; k3 , a3 ) = −igo e1 ·k23 (2π)26δ 26 (

a1 a2 a3 i ki )Tr λ [λ , λ ] ,

(6.5.12)

where kij ≡ ki − kj . This is again independent of the vertex operator positions. The s = 0 pole in the four-tachyon amplitude no longer vanishes. The terms that canceled now have the Chan–Paton factors in diﬀerent orders, so the pole is proportional to

Tr [λa1 , λa2 ] [λa3 , λa4 ] .

(6.5.13)

Relating the coeﬃcient of this pole to the amplitude (6.5.12) by unitarity, one obtains go = (2α )−1/2 go .

(6.5.14)

This is the same relative normalization as from the state–operator mapping: there is only one independent coupling constant.

188

6 Tree-level amplitudes

For the three-gauge-boson coupling, a similar calculation gives SD2 (k1 , a1 , e1 ; k2 , a2 , e2 ; k3 , a3 , e3 ) = igo (2π)26δ 26 (

i ki ) e1 ·k23 e2 ·e3 + e2 ·k31 e3 ·e1 + e3 ·k12 e1 ·e2

α + e1 ·k23 e2 ·k31 e3 ·k12 Tr λa1 [λa2 , λa3 ] . 2

(6.5.15)

Up to ﬁrst order in the momenta, the amplitudes we have found are reproduced by the spacetime action 1 S = 2 go

1 1 21/2 d x − Tr (Dµ ϕDµ ϕ) + Tr (ϕ2 ) + 1/2 Tr (ϕ3 ) 2 2α 3α 1 − Tr (Fµν F µν ) , (6.5.16) 4 26

where the tachyon ﬁeld ϕ and the Yang–Mills vector potential Aµ are written as n × n matrices, e. g. Aµ = Aaµ λa . Also, Dµ ϕ = ∂µ ϕ − i[Aµ , ϕ] and Fµν = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ]. This is the action for a U(n) gauge ﬁeld coupled to a scalar in the adjoint representation. Adding in the Chan–Paton factors has produced just the gauge-invariant expressions needed. The gauge invariance is automatic because the decoupling of unphysical states is guaranteed in string perturbation theory, as we will develop further. At momenta k small compared to the string scale, the only open string states are the massless gauge bosons. As discussed in section 3.7, it is in this limit that the physics should reduce to an eﬀective ﬁeld theory of the massless states. We have therefore been somewhat illogical in including the tachyon in the action (6.5.16); we did so for illustration, but now let us focus on the gauge bosons. The four-gauge-boson amplitude has a form analogous to the Veneziano amplitude but with additional structure from the polarization tensors. Expanding it in powers of α k 2 , the ﬁrst term, which survives in the zero-slope limit α → 0, is the sum of pole terms in s, t, and u plus a constant. Consistency guarantees that it is precisely the four-gauge-boson amplitude obtained in ﬁeld theory from the Yang–Mills Lagrangian Fµν F µν . Note, however, the term of order α k 3 in the three-gauge-boson amplitude (6.5.15). This implies a higher derivative term 2iα − 2 Tr (Fµ ν Fν ω Fω µ ) (6.5.17) 3go in the Lagrangian. Similarly, expanding the four-point amplitude reveals an inﬁnite sum of higher order interactions (beyond (6.5.17), these do not contribute to the three-gauge-boson amplitude for kinematic reasons). String loop amplitudes also reduce, in the low energy region, to the loops

6.5 Chan–Paton factors and gauge interactions

189

obtained from the eﬀective Lagrangian. By the usual logic of eﬀective actions, the higher derivative terms are less important at low energy. The scale where they become important, α k 2 ≈ 1, is just where new physics (massive string states) appears and the eﬀective action is no longer applicable. If we have a cutoﬀ, why do we need renormalization theory? Renormalization theory still has content and in fact this is its real interpretation: it means that low energy physics is independent of the details of the high energy theory, except for the parameters in the eﬀective Lagrangian. This is a mixed blessing: it means that we can use ordinary quantum ﬁeld theory to make predictions at accelerator energies without knowing the form of the Planck scale theory, but it also means that we cannot probe the Planck scale theory with physics at particle accelerators. The string spectrum and amplitudes have an obvious global U(n) symmetry, λa → Uλa U † ,

(6.5.18)

which leaves the Chan–Paton traces and the norms of states invariant. From the detailed form of the amplitudes, we have learned that this is actually a local symmetry in spacetime. We will see that this promotion of a global world-sheet symmetry to a local spacetime symmetry is a rather general phenomenon in string theory. All open string states transform as the n × n adjoint representation under the U(n) symmetry. Incidentally, U(n) is not a simple Lie algebra: U(n) = S U(n) × U(1). The U(1) gauge bosons, λij = δij /n1/2 , decouple from the amplitudes (6.5.12) and (6.5.15). The adjoint representation of U(1) is trivial, so all string states are neutral under the U(1) symmetry.

The unoriented string It is interesting to generalize to the unoriented string. Consider ﬁrst the theory without Chan–Paton factors. In addition to M¨ obius invariance, the µ X CFT on the sphere or disk is invariant under the orientation-reversing symmetry σ 1 → π − σ 1 in the open string or σ 1 → 2π − σ 1 in the closed string. This world-sheet parity symmetry is generated by an operator Ω. From the mode expansions it follows that Ωαµn Ω−1 = (−1)n αµn

(6.5.19)

Ωαµn Ω−1 = ˜αµn

(6.5.20)

in the open string and

190

6 Tree-level amplitudes

in the closed string. The symmetry extends to the ghosts as well, but to avoid distraction we will not discuss them explicitly, since they only contribute a ﬁxed factor to the tree-level amplitudes. The tachyon vertex operators are even under world-sheet parity in either the closed or open string (this is obvious for the integrated vertex operators, without ghosts; the ﬁxed operators must then transform in the same way), determining the sign of the operator. All states can then be classiﬁed by their parity eigenvalue ω = ±1. The relation (6.5.19) in the open string implies that Ω|N; k = ωN |N; k ,

2

ωN = (−1)1+α m .

(6.5.21)

World-sheet parity is multiplicatively conserved. For example, the threetachyon amplitude is nonzero, consistent with (+1)3 = 1. On the other hand, the massless vector has ω = −1 and so we would expect the vector– tachyon–tachyon amplitude (6.5.10) and three-vector amplitude (6.5.15) to vanish in the absence of Chan–Paton degrees of freedom, as indeed they do (the λa are replaced by 1 and the commutators vanish). The diﬀerent cyclic orderings, related to one another by world-sheet parity, cancel. Given a consistent oriented string theory, we can make a new unoriented string theory by restricting the spectrum to the states of ω = +1. States with odd α m2 remain, while states with even α m2 , including the photon, are absent. The conservation of ω guarantees that if all external states have ω = +1, then the intermediate states in tree-level amplitudes will also have ω = +1. Unitarity of the unoriented theory thus follows from that of the oriented theory, at least at tree level (and in fact to all orders, as we will outline in chapter 9). The main point of interest in the unoriented theory is the treatment of the Chan–Paton factors. Since we have identiﬁed these with the respective endpoints of the open string, world-sheet parity must reverse them, Ω|N; k; ij = ωN |N; k; ji .

(6.5.22)

Again, this is a symmetry of all amplitudes in the oriented theory. To form the unoriented theory, we again restrict the spectrum to world-sheet parity eigenvalue ω = +1. Take a basis for λaij in which each matrix is either symmetric, sa = +1, or antisymmetric, sa = −1. Then Ω|N; k; a = ωN sa |N; k; a .

(6.5.23)

The world-sheet parity eigenvalue is ω = ωN sa , and the unoriented spectrum is α m2 even: λa antisymmetric , α m2 odd: λa symmetric .

(6.5.24a) (6.5.24b)

6.5 Chan–Paton factors and gauge interactions

191

For the massless gauge bosons the Chan–Paton factors are the n × n antisymmetric matrices and so the gauge group is S O(n). The states at even mass levels transform as the adjoint representation of the orthogonal group S O(n), and the states at odd mass levels transform as the traceless symmetric tensor plus singlet representations. The oriented theory has a larger set of orientation-reversing symmetries, obtained as a combination of Ω and a U(n) rotation, Ωγ |N; k; ij = ωN γjj |N; k; j i γi−1 i .

(6.5.25)

We can form more general unoriented theories by restricting the spectrum to ωγ = +1, which is again consistent with the interactions. Acting twice with Ωγ gives Ω2γ |N; k; ij = [(γ T )−1 γ]ii |N; k; i j (γ −1 γ T )j j .

(6.5.26)

We will insist that Ω2γ = 1 for reasons to be explained below. This then implies that γ T = ±γ .

(6.5.27)

That is, γ is symmetric or antisymmetric. A general change of Chan–Paton basis, |N; k; ij = Uii−1 |N; k; i j Uj j ,

(6.5.28)

γ = U T γU .

(6.5.29)

transforms γ to

In the symmetric case, it is always possible to ﬁnd a basis such that γ = 1, giving the theory already considered above. In the antisymmetric case there is a basis in which 0 I . (6.5.30) γ=M≡i −I 0 Here I is the k × k identity matrix and n = 2k must be even because γ is an invertible antisymmetric matrix. We take a basis for the Chan– Paton wavefunctions such that M(λa )T M = sa λa with sa = ±1. Then the world-sheet parity eigenvalue is ωγ = ωN sa , and the unoriented spectrum is α m2 even: M(λa )T M = −λa , α m2 odd: M(λa )T M = +λa .

(6.5.31a) (6.5.31b)

At the even mass levels, including the gauge bosons, this deﬁnes the adjoint representation of the symplectic group S p(k). The argument that we must have Ω2γ = 1 to construct the unoriented theory is as follows. Since Ω2 = 1, it must be that Ω2γ acts only on the

192

6 Tree-level amplitudes

Chan–Paton states, not the oscillators. In fact, from eq. (6.5.26), it acts on the Chan–Paton wavefunction as λ → (γ T )−1 γλγ −1 γ T = λ ,

(6.5.32)

where the last equality must hold in the unoriented theory, since all states in this theory are invariant under Ωγ and so also under Ω2γ . Now we assert that the allowed Chan–Paton wavefunctions must form a complete set. The point is that two open strings, by the splitting–joining interaction of ﬁgure 3.4(c), can exchange endpoints. In this way one can get to a complete set, that is, to any Chan–Paton state |ij (this argument is slightly heuristic, but true). By Schur’s lemma, if eq. (6.5.32) holds for a complete set, then γ −1 γ T = 1 and so Ω2γ = 1. We might try to obtain other gauge groups by taking diﬀerent sets of λa . In fact, the oriented U(n) and unoriented S O(n) and S p(k) theories constructed above are the only possibilities. Generalization of the completeness argument in eqs. (6.5.7)–(6.5.9) shows these to be the most general solutions of the unitarity conditions. In particular, exceptional Lie algebras, which are of interest in grand uniﬁcation and which will play a major role in volume two, cannot be obtained with Chan–Paton factors (in perturbation theory). In closed string theory there is another mechanism that gives rise to gauge bosons, and it allows other groups. 6.6

Closed string tree amplitudes

The discussion of closed string amplitudes is parallel to the above. The amplitude for three closed string tachyons is gc3 e−2λ

SS2 (k1 ; k2 ; k3 ) =

3

iki ·X

:˜cce

(zi , ¯zi ) :

.

i=1

(6.6.1)

S2

In this case the CKG P S L(2, C) (the M¨ obius group) can be used to ﬁx the three vertex operators to arbitrary positions z1,2,3 . Taking the expectation values from section 6.2, the result is again independent of the vertex operator positions, SS2 (k1 ; k2 ; k3 ) = igc3 CS2 (2π)26δ 26 (

i ki )

(6.6.2)

,

where CS2 = e−2λ CSX2 CSg2 . For four closed string tachyons,

SS2 (k1 ; k2 ; k3 ; k4 ) =

gc4 e−2λ

C

2

d z4

3 i=1

iki ·X

:˜cce

ik4 ·X

(zi , ¯zi ) : : e

(z4 , ¯z4 ) :

, S2

(6.6.3)

193

6.6 Closed string tree amplitudes

where the integral runs over the complex plane C. Evaluating the expectation value and setting z1 = 0, z2 = 1, z3 = ∞, this becomes SS2 (k1 ; k2 ; k3 ; k4 ) = igc4 CS2 (2π)26δ 26 ( where

J(s, t, u) = C

i ki )J(s, t, u)

(6.6.4)

,

d2 z4 |z4 |−α u/2−4 |1 − z4 |−α t/2−4 .

(6.6.5)

Here s + t + u = −16/α , but we indicate the dependence of J on all three variables to emphasize its symmetry among them. This amplitude converges when s, t, u < −4/α . It has poles in the variable u from z4 → 0, in the variable t from z4 → 1, and in the variable s from z4 → ∞. The poles are at the values α s , α t , α u = −4, 0, 4, 8, . . . ,

(6.6.6)

which are the masses-squared of the closed string states. The pole at α s = −4 is 8πig 4 CS 4 (6.6.7) igc CS2 d2 z4 |z4 |α s/2 ∼ − c 2 . αs+4 |z4 |>1/ Unitarity gives CS2 =

8π , α gc2

(6.6.8)

and so 8πigc (2π)26δ 26 ( i ki ) . (6.6.9) α Like the Veneziano amplitude, the amplitude for four closed string tachyons can be expressed in terms of gamma functions (exercise 6.10): SS2 (k1 ; k2 ; k3 ) =

8πigc2 (2π)26δ 26 ( α where αc (x) = 1 + α x/4 and SS2 (k1 ; k2 ; k3 ; k4 ) =

C(a, b) = C

= 2π

i ki )C(−αc (t), −αc (u))

,

(6.6.10)

d2 z |z|2a−2 |1 − z|2b−2 Γ(a)Γ(b)Γ(c) , Γ(a + b)Γ(a + c)Γ(b + c)

a + b + c = 1 . (6.6.11)

This is the Virasoro–Shapiro amplitude. There is just a single term, with poles in the s-, t-, and u-channels coming from the gamma functions in the numerator. Like the Veneziano amplitude, the Virasoro–Shapiro amplitude has Regge behavior in the Regge limit, SS2 (k1 ; k2 ; k3 ; k4 ) ∝ s2αc (t)

Γ(−αc (t)) , Γ(1 + αc (t))

(6.6.12)

194

6 Tree-level amplitudes

and exponential behavior in the hard scattering limit,

α SS2 (k1 ; k2 ; k3 ; k4 ) ∝ exp − (s ln sα + t ln tα + u ln uα ) . (6.6.13) 2 On the sphere, the amplitude for a massless closed string and two closed string tachyons is

¯ ν eik1 ·X (z1 , ¯z1 ) : SS2 (k1 , e1 ; k2 ; k3 ) = gc2 gc e−2λ e1µν :˜cc∂X µ ∂X :˜cceik2 ·X (z2 , ¯z2 ) : :˜cceik3 ·X (z3 , ¯z3 ) :

S2

πiα µ ν =− k23 (2π)26δ 26 ( i ki ) , (6.6.14) g e1µν k23 2 c where e1µν eµν 1 = 1. Expanding the Virasoro–Shapiro amplitude (6.6.10) on the s = 0 pole and using unitarity determines 2 gc = gc , (6.6.15) α again in agreement with the state–operator mapping with overall constant gc . The amplitude (6.6.14) would be obtained in ﬁeld theory from the spacetime action S + S T , where S is the action (3.7.20) for the massless ﬁelds, and where 1 4 2 ˜ 26 1/2 −2Φ µν ST = − d x (−G) e G ∂µ T ∂ν T − T , (6.6.16) 2 α is the action for the closed string tachyon T . For example, the amplitude for a graviton of polarization eµν is obtained from this action by expanding ˜ µν = ηµν − 2κeµν eik·x . G

(6.6.17)

Note that this is the Einstein metric, whose action (3.7.25) is independent of the dilaton. The normalization of the ﬂuctuation is determined by that of the graviton kinetic term in the spacetime action. Speciﬁcally, if one ˜ µν − ηµν = −2κeµν f(x) with eµν eµν = 1, the eﬀective action for takes G f has the canonical normalization 12 for a real scalar. The ﬁeld theory amplitude matches the string result (6.6.14) and relates the normalization of the vertex operators to the gravitational coupling, κ = πα gc = 2πgc .

(6.6.18)

The amplitude for three massless closed strings is iκ SS2 (k1 , e1 ; k2 , e2 ; k3 , e3 ) = (2π)26δ 26 ( i ki )e1µν e2αβ e3γδ T µαγ T νβδ , 2 (6.6.19) where α µ α γ µ αγ γ µα α γµ η + k31 η + k12 η + k23 k31 k12 . (6.6.20) T µαγ = k23 8

6.6 Closed string tree amplitudes

195

The order k 2 terms in this amplitude correspond to the spacetime action (3.7.25), while the k 4 and k 6 terms come from a variety of higher derivative interactions, including terms quadratic and cubic in the spacetime curvature. These higher corrections to the action can also be determined by calculating higher loop corrections to the world-sheet beta functions (3.7.14). The tensor structure of the closed string amplitude (6.6.19) is just two copies of that in the open string amplitude (6.5.15), if one sets α = 2 in the closed string and α = 12 in the open. The same holds for the amplitude (6.6.14). This is a consequence of the factorization of free-ﬁeld expectation values on the sphere into holomorphic and antiholomorphic parts. A similar factorization holds for four or more closed strings before integration over the vertex operator positions. Further, by careful treatment of the contour of integration it is possible to ﬁnd relations between the integrated amplitudes. For the four-tachyon amplitudes the integrals above are related J(s, t, u, α ) = −2 sin παc (t) I(s, t, 4α ) I(t, u, 4α ) ,

(6.6.21)

after use of the gamma function identity Γ(x)Γ(1 − x) sin(πx) = π; we now indicate the explicit dependence of the integrals on α . For the general integral appearing in four-point closed string amplitudes there is a relation

C

d2 z z a−1+m1 ¯z a−1+n1 (1 − z)b−1+m2 (1 − ¯z )b−1+n2 = 2 sin[π(b + n2 )]B(a + m1 , b + m2 )B(b + n2 , 1 − a − b − n1 − n2 ) . m1 − n1 ∈ Z , m2 − n2 ∈ Z (6.6.22)

This implies a corresponding relation between four-point open and closed string amplitudes πigc2 α sin[παc (t)]Ao (s, t, 14 α , go )Ao (t, u, 14 α , go )∗ , go4 (6.6.23) where the open string amplitudes include just one of the six cyclic permutations, with poles in the indicated channels. Ac (s, t, u, α , gc ) =

Consistency In chapter 9 we will discuss the convergence and gauge invariance of tree-level amplitudes in a general way, but as an introduction we will now use the OPE to see how these work for the lowest levels. Consider the operator product

¯ : eik1 ·X(z1 ,¯z1 ) : : eik4 ·X(z4 ,¯z4 ) : = |z14 |α k1 ·k4 : 1 + iz14 k1 ·∂X + i¯z14 k1 · ∂X

¯ + . . . ei(k1 +k4 )·X (z4 , ¯z4 ) : . − z14 ¯z14 k1 ·∂X k1 · ∂X

(6.6.24)

196

6 Tree-level amplitudes

This appears in the amplitude with z14 integrated. The integral converges as z14 → 0 when α (6.6.25) (k1 + k4 )2 − 4 > −2 2 and has a pole at that point. The coeﬃcient of the pole is just the tachyon vertex operator. Thus, if a pair of tachyons in any amplitude has total momentum (k1 + k4 )2 = 4/α , there will be a pole proportional to the amplitude with one fewer tachyon as required by unitarity. A pole in momentum space corresponds to long distance in spacetime, so this is a process in which two tachyons scatter into one, which then propagates and interacts with the remaining particles. Carrying the OPE further, the O(z14 ) and O(¯z14 ) terms do not produce poles because the angular integration gives zero residue, the O(z14 ¯z14 ) term gives a massless pole, and so on. Now let us look a little more closely at the way the local spacetime symmetries are maintained in the string amplitudes. The various amplitudes we have calculated all vanish if any of the polarizations are of the form eµ = kµ , or eµν = ζµ kν + kµ ˜ζν with k·ζ = k· ˜ζ = 0. The corresponding spacetime actions are thus invariant under the Yang–Mills, coordinate, and antisymmetric tensor symmetries. As discussed in section 3.6, the vertex operator for a longitudinal polarization is the sum of a total derivative and a term that vanishes by the equations of motion. Upon integration, the total derivative vanishes but the equation of motion term might have a source at one of the other vertex operators in the path integral. The operator product (6.6.24) vanishes rapidly when k1 · k4 is large. Using this property, there will be for any pair of vertex operators a kinematic region in which all possible contact terms are suppressed. The amplitude for any null polarization then vanishes identically in this region, and since all amplitudes are analytic except for poles (and branch points at higher order) the amplitudes for null polarizations must vanish everywhere. We see that the singularities required by unitarity, as well as any possible divergences or violations of spacetime gauge invariance, arise from the limits z → 0, 1, ∞, where two vertex operators come together. For the sphere with four marked points (vertex operators) these are the boundaries of moduli space. The analytic continuation argument used here is known for historic reasons as the canceled propagator argument. α k1 · k4 =

Closed strings on D2 and RP2 The lowest order closed–open interactions come from the disk with both closed and open vertex operators. The low energy eﬀective action for these can be deduced from general considerations. With a trivial closed string

6.6 Closed string tree amplitudes background we found the usual gauge kinetic term, 1 − 2 d26 x Tr (Fµν F µν ) . 4go

197

(6.6.26)

Obviously the metric must couple to this in a covariant way. In addition, the coupling of the dilaton can be deduced. Recall that go2 ∝ eΦ0 , with Φ0 the expectation value of the dilaton. So we should replace this (inside the ˜ The action integral) with Φ = Φ0 + Φ. 1 ˜ − 2 d26 x (−G)1/2 e−Φ Tr (Fµν F µν ) (6.6.27) 4go thus incorporates all interactions not involving derivatives of the closed string ﬁelds; indices are now raised and lowered with Gµν . This action reﬂects the general principle that the eﬀective action from Euler number χ is weighted by gc−χ ∝ e−χΦ . The disk and projective plane also make a contribution to purely closed string interactions. The amplitudes for n closed strings are of order go−2 gcn ∼ gcn−1 , one power of gc higher than the sphere. Closed string loop amplitudes, to be considered in the next chapter, are of order gc2 times the sphere, because emitting and reabsorbing a closed string adds two factors of gc . Thus the disk and projective plane are ‘half-loop order.’ Of particular interest are the amplitudes on the disk and the projective plane with a single closed string vertex operator. Fixing the position of the vertex operator removes only two of the three CKVs, the residual gauge symmetry consisting of rotations about the vertex operator position. Thus we have to divide the amplitude by the volume of this residual CKG. We have not shown how to do this explicitly, but will work it out for the torus in the next chapter. For the disk with one closed string this is a ﬁnite factor and the result is nonzero. The amplitude is a numerical factor times gc go−2 , which in turn is a pure number, times powers of α as required by dimensional analysis. We will not work out these numerical factors here, but will obtain them in an indirect way in chapter 8. Thus there is an amplitude for a single closed string to appear from the vacuum, either through the disk or the projective plane, necessarily with zero momentum. Such an amplitude is known as a tadpole. In other words, the background closed string ﬁelds are corrected from their original values at order gc . Again we can write an eﬀective action, which is simply −Λ

˜

d26 x (−G)1/2 e−Φ ,

(6.6.28)

the Φ dependence deduced as above. This is a potential for the dilaton. We will consider it further in the next chapter. The amplitude for a single closed string on the sphere, on the other hand, is zero. The residual CKG is a noncompact subgroup of P S L(2, C) and so

198

6 Tree-level amplitudes

one has to divide by an inﬁnite volume. A nonzero result would have been a logical inconsistency, a zeroth order correction to the background ﬁelds. Similarly the amplitude for two closed strings on the sphere (a zeroth order correction to the mass) vanishes, as do the corresponding disk amplitudes, one or two open strings. The amplitudes with no vertex operators at all ˜ 0 in the are also meaningful — they just calculate the term of order Φ Taylor expansion of the action (6.6.28). The disk amplitude with no vertex operators is thus nonvanishing; this requires a somewhat formal treatment of the conformal Killing volume. 6.7

General results

In this section we obtain some general results concerning CFT on the sphere and disk. M¨obius invariance We have seen that the sphere has a group of globally deﬁned conformal transformations, the M¨ obius group P S L(2, C), z =

αz + β γz + δ

(6.7.1)

for complex α, β, γ, δ with αδ − βγ = 1. This is the most general conformal transformation that is one-to-one on all of S2 , the complex z-plane plus the point at inﬁnity. Expectation values must be invariant under any M¨ obius transformation:

Ai (z1 , ¯z1 ) . . . Ak (zn , ¯zn ) S2 = Ai (z1 , ¯z1 ) . . . Ak (zn , ¯zn )

S2

.

(6.7.2)

We will consider the consequences of this symmetry for expectation values with one, two, three, or four local operators. For a single operator of weight (hi , ˜hi ), the rescaling plus rotation z = γz gives

Ai (0, 0) S2 = Ai (0, 0)

˜

S2

= γ −hi ¯γ −hi Ai (0, 0) S2 .

(6.7.3)

The one-point function therefore vanishes unless hi = ˜hi = 0. This is another way to see that the one-point string amplitude vanishes on the sphere, because the matter factor is the expectation value of a (1, 1) operator. For n = 2, we can use a translation plus z = γz to bring any pair of operators to the points 0 and 1, giving −h −hj −˜ h −˜ h ¯z12 i j

Ai (z1 , ¯z1 )Aj (z2 , ¯z2 ) S2 = z12 i

Ai (1, 1)Aj (0, 0) S2 ,

(6.7.4)

199

6.7 General results

so the position dependence is completely determined. Single-valuedness implies that Ji + Jj ∈ Z, where Ji = hi − ˜hi . There is a further constraint on the two-point function from the conformal transformation z = z + (z − z1 )(z − z2 ) + O(2 ), which leaves z1 and z2 ﬁxed. For general operators this is complicated, but for tensor ﬁelds Op and Oq it simply implies that

Op (z1 , ¯z1 )Oq (z2 , ¯z2 ) S2 = 0

unless hp = hq , ˜hp = ˜hq .

(6.7.5)

Any three points z1,2,3 can be brought to given positions by a M¨ obius transformation. For n ≥ 3, M¨ obius invariance therefore reduces the expectation value from a function of n complex variables to a function of n − 3 complex variables. Again the result takes a simple form only for tensor ﬁelds. For example, for three tensor ﬁelds one ﬁnds 3

Opi (zi , ¯zi )

i=1

= Cp1 p2 p3

3 h−2(˜ hi + ˜ h−2(hi +hj ) ˜ hj )

¯zij

zij

(6.7.6)

,

i,j=1 i<j

S2

where Cp1 p2 p3 is independent of position and h = h1 + h2 + h3 . For four primary ﬁelds, 4 i=1

˜

Opi (zi , ¯zi )

= Cp1 p2 p3 p4 (zc , ¯zc ) (z12 z34 )h (¯z12 ¯z34 )h ×

4 −hi −hj −˜ hi − ˜ hj

zij

i,j=1 i<j

S2

¯zij

,

(6.7.7)

hi , and zc = z12 z34 /z13 z24 is the M¨ where h = i hi , ˜h = i ˜ obius invariant cross-ratio. The function Cp1 p2 p3 p4 (zc , ¯zc ) is not determined by conformal invariance, so we are reduced from a function of four variables to an unknown function of one variable. On the disk represented as the upper half-plane, only the M¨ obius transformations with α, β, γ, δ real remain, forming the group P S L(2, R). The extension of the above is left as an exercise. One learns much more by considering the full conformal algebra. We will see in chapter 15 that it determines all expectation values in terms of those of the tensor ﬁelds. Path integrals and matrix elements The path integrals we have considered can be related to operator expressions. Consider the path integral on the sphere with two operators, one at the origin and one at inﬁnity:

Ai (∞, ∞)Aj (0, 0)

S2

.

(6.7.8)

The prime indicates the u-frame, which we have to take for the operator at inﬁnity; by a slight abuse of notation we still give the position in terms of z. Using the state–operator mapping we can replace the disk |z| < 1 containing Aj at z = 0 by the state ΨAj on the circle |z| = 1. We can

200

6 Tree-level amplitudes

also replace the disk |z| > 1 (|u| < 1) containing Ai at u = 0 by the state ΨAi on the circle |z| = 1. All that is left is the integral over the ﬁelds φb on the circle, so the expectation value (6.7.8) becomes

[dφb ]ΨAi [φΩ b ]ΨAj [φb ] ;

(6.7.9)

here φΩ b (σ) = φb (2π − σ), arising from the mapping zu = 1. This convolution of wavefunctions resembles an inner product, so we deﬁne

i|j =

Ai (∞, ∞)Aj (0, 0)

S2

(6.7.10)

.

This is essentially the inner product introduced by Zamolodchikov. For tensor operators in unitary theories, a M¨ obius transformation shows that this is the same as the coeﬃcient of 1 in the OPE, Oi (z, ¯z )Oj (0, 0) =

i|j

˜ ˜ z hi +hj ¯z hi +hj

1 S 2

+ ... .

(6.7.11)

We have abbreviated |Ai to |i . The

| product is not the same as the quantum mechanical inner product | introduced in chapter 4. The latter is Hermitean, whereas the former includes no complex conjugation and so is bilinear, up to a sign if i and j are anticommuting. That is,

i|j = ±

j|i ,

(6.7.12)

since all we have done is to interchange the two operators and rename z ↔ u. There is a simple relation between the two inner products, which we will develop later. We will sometimes write Gij for

i|j , and Gij for the inverse matrix, where i, j run over a complete set. The matrices Gij and Gij will be used to raise and lower indices, Ai = Gij Aj , Ai = Gij Aj . Operators in the path integral translate in the usual way into operators in Hilbert space. For example, ˆ k (1, 1)|j , A (∞, ∞)Ak (1, 1)Aj (0, 0) =

i|A (6.7.13) i

S2

where we reintroduce the hat to emphasize that we are in a Hilbert space formalism. Using the OPE, the left-hand side becomes

clkj Ai (∞, ∞)Al (0, 0)

S2

= cikj .

(6.7.14)

l

Thus the three-point expectation values on the sphere, the OPE coeﬃcients with all indices lowered, and the matrix elements of general local operators are all the same thing. By a M¨ obius transformation (rescaling of z) we have also

Ai (∞, ∞)Ak (z1 , ¯z1 )Aj (0, 0)

S2

˜ −˜ h −hk −hj h h −˜ h ¯z1 i k j cikj

= z1 i

.

(6.7.15)

201

6.7 General results

The four-point function translates into an operator expression ˆ k (z1 , ¯z1 )A ˆ l (z2 , ¯z2 ) ]|j , A (∞, ∞)Ak (z1 , ¯z1 )Al (z2 , ¯z2 )Aj (0, 0) =

i|T[ A i

S2

(6.7.16) where T denotes radial ordering. Let |z1 | > |z2 | and insert a complete set of states, 1 = |m Gmn

n| .

(6.7.17)

The four-point amplitude (6.7.16) becomes h −h −h ˜h −˜h −˜h hm −hl −hj ˜hm −˜hl −˜hj m m i i k k

¯z1

z1

¯z2

z2

cikm cmlj .

(6.7.18)

m

Thus, the operator product coeﬃcients determine not only the three-point expectation values on the sphere, but also the four-point and, by the same construction, arbitrary n-point amplitudes. For |z1 − z2 | > |z1 |, which overlaps the region |z1 | > |z2 | where the expansion (6.7.18) is valid, we can translate Ak to the origin and give a similar expansion in terms of cilm cmkj . The equality of these two expansions is associativity of the OPE, ﬁgure 2.8. Operator calculations The Hilbert space expressions give us one more way to calculate expectations values. We take as an example the case of four exponential operators

: eik4 ·X(∞,∞) : : eik1 ·X(z1 ,¯z1 ) : : eik2 ·X(z2 ,¯z2 ) : : eik3 ·X(0,0) : ik1 ·X1 ◦

=

0; k4 |T[ e ◦ ◦

◦ ◦ ◦

ik2 ·X2 ◦

e

◦

S2

]|0; k3 .

(6.7.19)

We have used the result (2.7.11) that : : ordered operators are the same as ◦◦ ◦◦ ordered operators for this CFT. We have abbreviated X µ (zi , ¯zi ) as Xiµ and to avoid clutter are omitting the hats on operators. By deﬁnition ◦ ◦

where

eik·X ◦◦ = eik·XC eik·XA ,

1/2 ∞ α 1

XCµ (z, ¯z )

= x −i

XAµ (z, ¯z )

α α = −i pµ ln |z|2 + i 2 2

µ

2

m m=1

(6.7.20)

(αµ−m z m + ˜αµ−m ¯z m ) , 1/2 ∞ 1 αµm

˜αµm + m z m ¯z m m=1

(6.7.21a)

.

(6.7.21b)

For |z1 | > |z2 | the matrix element (6.7.19) becomes

0; k4 |eik1 ·X1C eik1 ·X1A eik2 ·X2C eik2 ·X2A |0; k3 .

(6.7.22)

202

6 Tree-level amplitudes

To evaluate this use the Campbell–Baker–Hausdorﬀ (CBH) formula eik1 ·X1A eik2 ·X2C = eik2 ·X2C eik1 ·X1A e−[ k1 ·X1A , k2 ·X2C ] = eik2 ·X2C eik1 ·X1A |z12 |α k1 ·k2 .

(6.7.23)

The expectation value (6.7.22) becomes

|z12 |α k1 ·k2

0; k4 |eik1 ·X1C +ik2 ·X2C eik1 ·X1A +ik2 ·X2A |0; k3

= |z12 |α k1 ·k2

0; k4 |ei(k1 +k2 )·x eα (k1 ln |z1 |+k2 ln |z2 |)·p |0; k3

= |z12 |α k1 ·k2 |z1 |α k1 ·k3 |z2 |α k2 ·k3

0; k1 + k2 + k4 |0; k3

= iCSX2 (2π)d δ d (

i ki )|z12 |

α k1 ·k2

|z1 |α k1 ·k3 |z2 |α k2 ·k3 ,

(6.7.24)

where we have used the two-point expectation value to normalize the last line. This is the familiar result (6.2.31), obtained by two other methods in 2 section 6.2, after one includes in the latter a factor |z4 |α k4 from the change of frame and takes z4 → ∞. All other free-ﬁeld results can be obtained by this same oscillator method. Relation between inner products Nondegenerate bilinear and Hermitean inner products can always be related to one another by an appropriate antilinear operation on the bra. Let us consider an example. From the free-ﬁeld expectation values, we have

0; k|0; l = iCSX2 (2π)d δ d (k + l) ,

(6.7.25)

using the fact that |0; k maps to eik·X . Compare this with the inner product from the X µ CFT,

0; k|0; l = (2π)d δ d (l − k) .

(6.7.26)

These diﬀer only by k → −k from conjugating eik·X , and normalization,

0; k| = iCSX2 0; −k| .

(6.7.27)

For more general operators, there is a natural notion of conjugation in CFT. In Euclidean quantum mechanics, Hermitean conjugation inverts Euclidean time, eq. (A.1.37), so the natural operation of Euclidean conjugation is conjugation × time-reversal: an operator that is Hermitean in Minkowski space is also Hermitean under this combined operation. In CFT, we make the same deﬁnition, but also must include a time-reversal on the conformal frame, A(p) = A (p )† .

(6.7.28)

Here p and p are related by radial time-reversal, z = ¯z −1 , and the unprimed operator is in the z-frame and the primed operator in the u-

203

6.7 General results

frame. To see how this works, consider a holomorphic operator of weight h, whose Laurent expansion is O(z) = ih

∞

On . n+h n=−∞ z

(6.7.29)

Its simple adjoint is O(z)† = i−h

∞

O†n . z n+h n=−∞ ¯

(6.7.30)

Then the Euclidean adjoint is O(z) = i−h (−z −2 )h

∞ n=−∞

O†n

z −n−h

= ih

∞ O†−n n=−∞

z n+h

.

(6.7.31)

For an operator that is Hermitean in Minkowski time, O†−n = On , and so this operator is also Hermitean3 under O. The Euclidean adjoint (6.7.28) conjugates all explicit factors of i, but leaves z and ¯z indices unchanged. This is its whole eﬀect, other than an overall factor (−1)Na (Na −1)/2 from reversing the order of anticommuting ﬁelds; here Na is the total number of anticommuting ﬁelds in the operator. This is the natural conformally invariant operation of conjugation, so it must be that (6.7.32)

Ai | = K Ai | for some constant K. For a direct demonstration, the best we have come up with is

i | j ≡

1|Ai (∞, ∞)|j = K 1|Ai (∞, ∞)|j

= K 1|Ai (0, 0)† |j = K j|Ai (0, 0)|1 ∗ = K j|i ∗ = K i|j .

(6.7.33)

The only step here that is not either a deﬁnition or obvious is the assumption of proportionality,

1| = K 1|. This must hold because |1 is the unique S L(2,C)-invariant state. For the X CFT we have seen that K X = iCSX2 . For the ghost CFT, the Laurent expansion of the amplitude (6.3.4) gives

0|˜c0 c0 |0 = −CSg2 , where |0 = ˜c1 c1 |1 . The Hermitean inner product was deﬁned in eq. (4.3.18) as

0|˜c0 c0 |0 = i, so K g = iCSg2 . Except for the is, which are absent in a unitary CFT, one can set K = 1 by adding an Euler number term to the action. In a Hermitean basis of operators, the two inner products are then identical, and the distinction 3

Because its Laurent expansion has no factor of i, the Faddeev–Popov c ghost is actually anti-Hermitean under the Euclidean adjoint. This is due to an inconvenient conﬂict between conventions, and has no signiﬁcance.

204

6 Tree-level amplitudes

between them is often ignored. Notice, however, that a vertex operator of deﬁnite nonzero momentum cannot be Hermitean. Again, this can all be extended to the open string, with the disk in place of the sphere. Exercises 6.1 Verify that the expectation value (6.2.31) is smooth in the u-patch. 6.2 For the linear dilaton CFT, Φ = Vµ X µ , the X µ zero-mode path integral diverges. This reﬂects the fact that the coupling is diverging in some direction. Consider therefore imaginary Φ = ibµ X µ ; this is unphysical but has technical applications (one could instead take complex momenta). (a) Generalize the calculation that gave (6.2.17) to this case. Show that in the end the only changes are in the Weyl dependence (reﬂecting a change in the dimension of the exponential) and in the spacetime momentum conservation. (b) Generalize the calculation of the same quantity using holomorphicity. Show that the result is the same as in the regular X µ CFT but with an extra background charge operator exp(−2ib · X) taken to inﬁnity. 6.3 For one b and four c ﬁelds, show that expressions (6.3.5) and (6.3.8) are equal. Ignore the antiholomorphic ﬁelds. 6.4 (a) Write the amplitude for n open string tachyons on the disk as a generalization of eq. (6.4.5). (b) Show by a change of variables that it is independent of the positions of the ﬁxed vertex operators. 6.5 (a) Show that the residue of the pole in I(s, t) at α s = J − 1 is a polynomial in u − t of degree J, corresponding to intermediate particles of spin up to J. (b) Consider the amplitude I(s, t) without assuming D = 26. In the centerof-mass frame, write the residue of the pole at α s = 1 in terms of J k3k k3l , k1i k1j Pij,kl

where P 0 projects onto spin 0 (the unit matrix) and P 2 projects onto spin 2 (traceless symmetric tensors). Show that the coeﬃcient of P 0 is positive for D < 26, zero at D = 26, and negative at D > 26. Compare with the expected string spectrum at this level, and with the discussion of this level in section 4.1. 6.6 (a) In the hard scattering limit, evaluate the integral in the Veneziano amplitude (6.4.5) using the saddle point approximation. Take the naive saddle point; the justiﬁcation requires consideration of the analytic continuations needed to deﬁne the integral. (b) Do the same for the Virasoro–Shapiro amplitude.

Exercises

205

6.7 (a) Derive the amplitude (6.5.12). (b) Verify the relation (6.5.14) between vertex operator normalizations. Show that this agrees with the state–operator mapping. 6.8 Derive the three-gauge-boson amplitude (6.5.15). 6.9 (a) Obtain the four-gauge-boson amplitude. To reduce the rather extensive algebra, consider only those terms in which the polarizations appear as e1 ·e2 e3 ·e4 . (b) Compare the behavior at small α ki · kj with the same amplitude in Yang–Mills theory. 6.10 Carry out the integral in eq. (6.6.5) to obtain the Virasoro–Shapiro amplitude (6.6.10). The relation |z|

−2a

1 = Γ(a)

∞

dt ta−1 exp(−tz¯z )

0

is useful. (Reference: Green, Schwarz, & Witten (1987), section 7.2; note that their d2 z is 12 times ours.) 6.11 (a) Verify the amplitude (6.6.14). (b) Verify the relation (6.6.15) between vertex operator normalizations. (c) Complete the ﬁeld theory calculation and verify the relation (6.6.18) between gc and the gravitational coupling. 6.12 Calculate the disk amplitude with one closed string tachyon and two open string tachyons. 6.13 (a) Find the P S L(2, C) transformation that takes three given points z1,2,3 into chosen positions zˆ 1,2,3 . (b) Verify the M¨ obius invariance results (6.7.3)–(6.7.7). Show that to derive ˜ 1 annihilate the operators. (6.7.5) it is suﬃcient that L1 and L 6.14 (a) Find the P S L(2, R) transformation that takes three given points y1,2,3 into chosen positions yˆ 1,2,3 . (b) Generalize the M¨ obius invariance results (6.7.3)–(6.7.7) to operators on the boundary of a disk. (c) Find similar results for the disk with one interior, two interior, and one interior and one boundary operators. 6.15 (a) Generalize the operator calculation (6.7.24) to a product of n exponentials. (b) Generalize it to the expectation value (6.2.25).

7 One-loop amplitudes

After describing the relevant surfaces, we focus on the torus, ﬁrst on the CFTs on this surface and then on the string amplitudes. We then generalize to the open and unoriented theories. The most important issue to be understood is the absence of short-distance (UV) divergences. 7.1

Riemann surfaces

There are four Riemann surfaces with Euler number zero. The torus The torus T 2 , discussed in section 5.1, is the only closed oriented surface with Euler number zero. We describe it as the complex plane with metric ¯ and identiﬁcations ds2 = dwdw w∼ (7.1.1) = w + 2π ∼ = w + 2πτ . There are two moduli, the real and imaginary parts of τ = τ1 + iτ2 , and two CKVs, the translations. In terms of the real coordinates w = σ 1 + iσ 2 , (σ 1 , σ 2 ) ∼ (7.1.2) = (σ 1 + 2π, σ 2 ) ∼ = (σ 1 + 2πτ1 , σ 2 + 2πτ2 ) , so that one can think of the torus as a cylinder of circumference 2π and length 2πτ2 with the ends rotated by an angle of 2πτ1 and then sewn together. In terms of the coordinate z = exp(−iw), the identiﬁcation w ∼ = w + 2π is automatic, while w ∼ = w + 2πτ becomes z∼ (7.1.3) = z exp(−2πiτ) . A fundamental region is the annulus 1 ≤ |z| ≤ exp(2πτ2 ) . 206

(7.1.4)

7.1 Riemann surfaces

207

The torus is formed by rotating the outer circle by 2πτ1 and then sewing the inner and outer circles together. We will use the w-coordinate unless otherwise noted. The cylinder (annulus) The cylinder C2 is the region 0 ≤ Re w ≤ π ,

w∼ = w + 2πit .

(7.1.5)

That is, it is a strip of width π and length 2πt, with the ends joined. There is a single modulus t, which runs over the whole range 0 < t < ∞. Unlike the torus there is no modular group, the long-cylinder limit t → 0 being quite diﬀerent from the long-strip limit t → ∞. There is a single CKV, the translation parallel to the boundary. The cylinder can be obtained from the torus with imaginary τ = it by identifying under the involution ¯ , w = −w

(7.1.6)

which is a reﬂection through the imaginary axis. The lines σ 1 = 0, π are ﬁxed by this reﬂection and so become boundaries. The other coordinate is periodically identiﬁed, (σ 1 , 0) ∼ = (σ 1 , 2πt) .

(7.1.7)

The Klein bottle The Klein bottle K2 can be regarded as the complex plane with identiﬁcation ¯ + 2πit w∼ = w + 2π ∼ = −w

(7.1.8)

(σ 1 , σ 2 ) ∼ = (σ 1 + 2π, σ 2 ) ∼ = (−σ 1 , σ 2 + 2πt) .

(7.1.9)

or

This is a cylinder of circumference 2π and length 2πt, with the ends sewn together after a parity-reversal Ω. The single modulus t runs over 0 < t < ∞, and there is no modular group. Translation in the σ 2 direction is the only CKV. The Klein bottle can be obtained from the torus with modulus τ = 2it by identifying under ¯ + 2πit . w = −w

(7.1.10)

The Klein bottle can be thought of as a sphere with two cross-caps, but we postpone this description until section 7.4.

208

7 One-loop amplitudes The M¨obius strip

The M¨ obius strip M2 is sewn from a strip with a twist by Ω, 0 ≤ Re w ≤ π ,

¯ + π + 2πit . w∼ = −w

(7.1.11)

Again the modulus t runs over 0 < t < ∞ and the only CKV is the σ 2 translation. It can be obtained from the torus with τ = 2it by identifying under the two involutions ¯ w = −w

and

w = w + π(2it + 1) .

(7.1.12)

The M¨ obius strip can be thought of as a disk with a cross-cap, but again we defer this description. 7.2

CFT on the torus Scalar correlators

As in the case of the sphere, we start with the Green’s function (6.2.7), satisfying 2¯ 1 ¯ w, w ¯ ) = −2πδ 2 (w − w ) + ∂∂G (w, w; . α 4πτ2

(7.2.1)

The Green’s function is periodic in both directions on the torus, and aside from the source and the background charge term it would be the sum of holomorphic and antiholomorphic functions. These properties identify it as being associated with the theta functions, whose properties are given at the end of this section. In particular we guess ¯ w, w ¯ ) ∼ − G (w, w;

w − w 2 α ln ϑ1 τ . 2 2π

(7.2.2)

The function ϑ1 vanishes linearly when its ﬁrst argument goes to zero, giving the correct behavior as w → w . Elsewhere the logarithm is the sum of holomorphic and antiholomorphic functions and so is annihilated ¯ However, this function is not quite doubly periodic owing to the by ∂∂. quasiperiodicity (7.2.32b) — it changes by −α [Im(w − w ) + πτ2 ] under w → w + 2πτ. Also, the background charge is missing. Both properties are easily corrected:

w − w 2 [Im(w − w )]2 α + k(τ, ¯τ) . ln ϑ1 τ + α 2 2π 4πτ2 (7.2.3) The function k(τ, ¯τ) is determined by orthogonality to X0 , but as in the case of the sphere it drops out due to spacetime momentum conservation. ¯ w, w ¯ ) = − G (w, w;

209

7.2 CFT on the torus

The expectation value of a product of vertex operators is given, in parallel to the earlier result (6.2.13) on the sphere, by n i=1

iki ·X(zi ,¯zi )

:e

= iCTX2 (τ)(2π)d δ d (

:

i ki )

T2

α ki ·kj 2π wij (Im wij )2 . × ϑ τ exp − ∂ν ϑ1 (0|τ) 1 2π 4πτ2

(7.2.4)

i<j

The factor 2π/∂ν ϑ1 comes from the renormalized self-contractions. The scalar partition function The overall normalization of the sphere was absorbed into the string coupling constant, but we cannot do this for the torus, in particular because there is a nontrivial τ-dependence. In fact, we will see that there is a great deal of physics in the amplitude with no vertex operators at all. Consider the path integral with no vertex operators, 1 T 2 (τ) ≡ Z(τ). We can think of the torus with modulus τ as formed by taking a ﬁeld theory on a circle, evolving for Euclidean time 2πτ2 , translating in σ 1 by 2πτ1 , and then identifying the ends. In operator language this gives a trace,

Z(τ) = Tr exp(2πiτ1 P − 2πτ2 H)

˜

= (q¯ q )−d/24 Tr q L0 q¯L0

(7.2.5)

.

˜ 0 generates translations of Here q = exp(2πiτ), the momentum P = L0 − L 1 1 ˜ σ , and the Hamiltonian H = L0 + L0 − 24 (c + ˜c) generates translations of σ 2 as in eq. (2.6.10). Such a trace, weighted by the exponential of the Hamiltonian and other conserved quantities, is termed a partition function as in statistical mechanics. The trace breaks up into a sum over occupation numbers Nµn and ˜ µn for each µ and n and an integral over momentum k µ . The Virasoro N generators similarly break up into a sum, and Z(τ) becomes q )−d/24 Vd (q¯

dd k exp(−πτ2 α k 2 ) d (2π) µ,n

∞

˜

q nNµn q¯nNµn .

(7.2.6)

˜ µn =0 Nµn ,N

normalizaThe factor of spacetime volume Vd comes from the continuum −d d d k. The various sums tion of the momentum, k becoming Vd (2π) are geometric, ∞ N=0

q nN = (1 − q n )−1 ,

(7.2.7)

210

7 One-loop amplitudes

and so we obtain Z(τ) = iVd ZX(τ)d ,

(7.2.8)

ZX(τ) = (4π 2 α τ2 )−1/2 |η(τ)|−2 .

(7.2.9)

where

Here η(τ) = q 1/24

∞

(1 − q n )

(7.2.10)

n=1

is the Dedekind eta function, discussed further at the end of the section. The i comes from the rotation k 0 → ik d needed to deﬁne the divergent integral. Also, the earlier CTX2 (τ) is ZX(τ)d . The result (7.2.8) is obviously invariant under τ → τ + 1. Invariance of ZX under τ → −1/τ follows from eq. (7.2.44), η(−1/τ) = (−iτ)1/2 η(τ), thus generating the full modular group. Similarly the expectation value with vertex operators is modular-covariant, taking into account the Weyl transformations of the operators. We have evaluated the path integral by translating into an operator expression, but as on the sphere there are two other methods. We leave to the reader the direct path integral evaluation. Fourier transforming in σ 1 , it just breaks up into a product of harmonic oscillator path integrals. The evaluation using holomorphicity is less direct. We include it for illustration because it is used at higher genus and for more complicated CFTs. The idea is to get a diﬀerential equation with respect to τ. Consider the torus with modulus τ and make a small change in the metric, δgww = ∗ . The new metric is ¯ + ∗ dw 2 + dw ¯2 ds2 = dwdw ¯ − w)] d[w ¯ + ∗ (w − w)] ¯ + O(2 ) . (7.2.11) = (1 + ∗ + ) d[w + (w ¯ with w = w + (w ¯ − w). That is, the metric is Weyl-equivalent to dw dw This has the periodicities w ∼ (7.2.12) = w + 2π(τ − 2iτ2 ) . = w + 2π ∼ This change in the metric is therefore equivalent to a change δτ = −2iτ2

(7.2.13)

in the modulus. Under a change in the metric, the change in the path integral is

1 ¯

d2 w δgw¯ w¯ Tww (w) + δgww Tw¯ w¯ (w) δZ(τ) = − 2π

= −2πi δτ Tww (0) − δ¯τ Tw¯ w¯ (0) .

(7.2.14)

211

7.2 CFT on the torus

In the second line we have used translation invariance to carry out the integral over w. All expectation values are understood to be on the torus with modulus τ. To get the expectation value of the energy-momentum tensor, we use the OPE ∂w X µ (w)∂w Xµ (0) = −

α d − α Tww (0) + O(w) . 2w 2

(7.2.15)

Now, ¯ w, w ¯ )|w =0 Z(τ)−1 ∂w X µ (w)∂w Xµ (0) = d∂w ∂w G (w, w; =

α d ϑ1 ∂w2 ϑ1 − ∂w ϑ1 ∂w ϑ1 α d + , 2 8πτ2 ϑ21

(7.2.16)

where the arguments of all the theta functions are (w/2π, τ). This does indeed have a double pole at w = 0. Carefully expanding both numerator and denominator to order w 2 (this is simpliﬁed by the fact that ϑ1 is odd), the order w 0 term in the expectation value (7.2.16) is α d α d ∂w3 ϑ1 + . 6 ∂w ϑ1 8πτ2

(7.2.17)

The OPE (7.2.15) then gives

d ∂3 ϑ1 (0|τ) d

Tww (0) = − w − 6 ∂w ϑ1 (0|τ) 8πτ2

Z(τ) ,

(7.2.18)

and the variation (7.2.14) gives the diﬀerential equation ∂τ ln Z(τ) =

πid ∂w3 ϑ1 (0|τ) id . + 3 ∂w ϑ1 (0|τ) 4τ2

(7.2.19)

To proceed further, we use the fact, which is easily veriﬁed from the inﬁnite sum, that the theta functions satisfy

∂w2 ϑ1

w w i τ = ∂τ ϑ1 τ 2π π 2π

.

(7.2.20)

Rewrite (7.2.19) as d id ∂τ ln Z(τ) = − ∂τ ln ∂w ϑ1 (0|τ) + . 3 4τ2

(7.2.21)

Together with the conjugate equation, this determines −d/2

Z(τ) = |∂w ϑ1 (0|τ)|−2d/3 τ2

(7.2.22)

up to a numerical coeﬃcient that must be ﬁxed by other means. Noting the relation (7.2.43), this is the same as the earlier result (7.2.8).

212

7 One-loop amplitudes The bc CFT

For the ghost CFT, the partition function is again determined easily in terms of a trace over states. There are two sets of raising operators b−n and c−n on both the right and left, but now these are anticommuting so the occupation numbers can only be zero or one — a single oscillator at mode n thus gives (1 + q n ). In addition there are four ground states, giving

˜

Tr exp(2πiτ1 P − 2πτ2 H) = (q¯ q )13/12 Tr q L0 q¯L0 = 4(q¯ q )1/12

∞

|1 + q n |4 .

(7.2.23)

n=1

Recall, however, that for anticommuting ﬁelds the simple trace corresponds to a path integral with antiperiodic boundary conditions in the time direction. In order to calculate the Faddeev–Popov determinant, the ghosts must have the same periodicity as the original coordinate transformations, which were periodic. Thus we need

Z(τ) = Tr (−1)F exp(2πiτ1 P − 2πτ2 H)

=0,

(7.2.24)

where (−1)F anticommutes with all the ghost ﬁelds. The trace vanishes because the states | ↓↓ and | ↑↑ have opposite ghost number mod 2 from | ↑↓ and | ↓↑ . From the path integral point of view, Z(τ) vanishes due to ghost zero modes from the moduli and CKVs. These must be saturated with appropriate insertions, the simplest nonvanishing amplitude (and the one we will need) being

¯ 3 )˜b(w ¯ 4) c(w1 )b(w2 )˜c(w

(7.2.25)

.

In the operator calculation we write each ﬁeld in terms of its mode expansion. Only the n = 0 terms contribute, the expectation values of the others vanishing as above. Then (7.2.25) becomes

Tr (−1)F c0 b0˜c0 ˜b0 exp(2πiτ1 P − 2πτ2 H) .

(7.2.26)

b0 projects onto the single ground state | ↑↑ , so the The operator c0 b0˜c0 ˜ result is (q¯ q )1/12

∞

|1 − q n |4 = |η(τ)|4 .

(7.2.27)

n=1

Note the sign change in the inﬁnite product, from the (−1)F in the trace. The result (7.2.27) is independent of the positions of the ghost ﬁelds because the CKV and quadratic diﬀerentials on the torus are constants.

7.2 CFT on the torus

213

Again, we do not try to keep track of the overall sign, setting it to be positive.

General CFTs For general CFTs, Z(τ) =

˜

q hi −c/24 q¯hi −˜c/24 (−1)Fi ,

(7.2.28)

i

where i runs over all states of the CFT, and Fi is again the world-sheet 1 (c − ˜c) fermion number. Invariance under τ → τ + 1 requires hi − ˜hi − 24 to be an integer. Looking at the unit operator, this requires c − ˜c to be a multiple of 24. For general operators one then has the requirement that the spin be an integer, hi − ˜hi ∈ Z .

(7.2.29)

Incidentally, CFTs for which c − ˜c is not a multiple of 24 are certainly of interest. These cannot be separately modular-invariant, but with appropriate projections can be combined into a modular-invariant theory. Invariance under τ → −1/τ puts a further constraint on the spectrum. Let us here note just one general consequence, which relates the states of large h to those of lowest h. Consider the partition function for τ = i as → 0. The convergence factor q = exp(−2π) is approaching unity, so the partition function is determined by the density of states at high weight. Using modular invariance, this is equal to the partition function at τ = i/. In the latter case, q = exp(−2π/) is going to zero, so the state of ˜ 0 = 0, lowest weight dominates the sum. This is the unit state, with L0 = L giving →0

Z(i) ≈ exp

π(c + ˜c) . 12

(7.2.30)

The density of states at high weight is thus determined by the central charge. This generalizes the free-boson result, where c = ˜c = d counts the number of free bosons.1

1

To be precise, the result (7.2.30) holds in compact unitary CFTs. In nonunitary theories there are usually operators of negative weight, whose contribution will dominate that of the unit operator. In the linear dilaton theory, the central charge depends on V µ but the density of states does not: here the exception occurs because of complications due to the noncompactness of X µ combined with lack of translation invariance.

214

7 One-loop amplitudes Theta functions

The basic theta function is ∞

ϑ(ν, τ) =

exp(πin2 τ + 2πinν) .

(7.2.31)

n=−∞

It has the periodicity properties ϑ(ν + 1, τ) = ϑ(ν, τ) , ϑ(ν + τ, τ) = exp(−πiτ − 2πiν)ϑ(ν, τ) ,

(7.2.32a) (7.2.32b)

and these in fact determine it up to normalization. Under modular transformations, ϑ(ν, τ + 1) = ϑ(ν + 1/2, τ) , ϑ(ν/τ, −1/τ) = (−iτ)

1/2

(7.2.33a) 2

exp(πiν /τ)ϑ(ν, τ) .

(7.2.33b)

The theta function has a unique zero, up to the periodicity (7.2.32), at ν = 12 (1 + τ). It can also be written as an inﬁnite product, ϑ(ν, τ) =

∞

(1 − q m )(1 + zq m−1/2 )(1 + z −1 q m−1/2 ) ,

(7.2.34)

m=1

in terms of q = exp(2πiτ),

z = exp(2πiν) .

(7.2.35)

We often need the asymptotic behavior of the theta function as q → 0 or as q → 1. The q → 0 behavior can be read immediately from either the inﬁnite sum or inﬁnite product form. The q → 1 behavior is not manifest in either form, because an inﬁnite number of terms contribute. It can be obtained from the τ → −1/τ modular transformation, which relates these two limits by q → exp(4π 2 / ln q). It is also useful to deﬁne the theta function with characteristics,

ϑ

a (ν, τ) = exp πia2 τ + 2πia(ν + b) ϑ(ν + aτ + b, τ) b

=

∞

exp πi(n + a)2 τ + 2πi(n + a)(ν + b) .

(7.2.36)

n=−∞

Other common notations are

ϑ00 (ν, τ) = ϑ3 (ν|τ) = ϑ

ϑ01 (ν, τ) = ϑ4 (ν|τ) = ϑ

∞

2 0 (ν, τ) = q n /2 z n , 0 n=−∞

(7.2.37a)

∞

2 0 (ν, τ) = (−1)n q n /2 z n , 1/2 n=−∞

(7.2.37b)

215

7.2 CFT on the torus

∞

2 1/2 (ν, τ) = q (n−1/2) /2 z n−1/2 , ϑ10 (ν, τ) = ϑ2 (ν|τ) = ϑ 0 n=−∞ (7.2.37c) 1/2 (ν, τ) ϑ11 (ν, τ) = −ϑ1 (ν|τ) = ϑ 1/2 ∞

= −i

(−1)n q (n−1/2)

2 /2

z n−1/2 .

(7.2.37d)

n=−∞

These have the product representations ϑ00 (ν, τ) = ϑ01 (ν, τ) =

∞

(1 − q m )(1 + zq m−1/2 )(1 + z −1 q m−1/2 ) ,

m=1 ∞

(1 − q m )(1 − zq m−1/2 )(1 − z −1 q m−1/2 ) ,

(7.2.38a) (7.2.38b)

m=1

ϑ10 (ν, τ) = 2 exp(πiτ/4) cos πν

∞

(1 − q m )(1 + zq m )(1 + z −1 q m ) ,

m=1

ϑ11 (ν, τ) = −2 exp(πiτ/4) sin πν

(7.2.38c)

∞

(1 − q m )(1 − zq m )(1 − z −1 q m ) .

m=1

(7.2.38d)

Their modular transformations are ϑ00 (ν, τ + 1) ϑ01 (ν, τ + 1) ϑ10 (ν, τ + 1) ϑ11 (ν, τ + 1)

= = = =

ϑ01 (ν, τ) , ϑ00 (ν, τ) , exp(πi/4)ϑ10 (ν, τ) , exp(πi/4)ϑ11 (ν, τ) ,

(7.2.39a) (7.2.39b) (7.2.39c) (7.2.39d)

and ϑ00 (ν/τ, −1/τ) = (−iτ)1/2 exp(πiν 2 /τ)ϑ00 (ν, τ) ,

(7.2.40a)

ϑ01 (ν/τ, −1/τ) = (−iτ)1/2 exp(πiν 2 /τ)ϑ10 (ν, τ) ,

(7.2.40b)

ϑ10 (ν/τ, −1/τ) = (−iτ)

(7.2.40c)

1/2

ϑ11 (ν/τ, −1/τ) = −i(−iτ)

2

exp(πiν /τ)ϑ01 (ν, τ) ,

1/2

2

exp(πiν /τ)ϑ11 (ν, τ) .

(7.2.40d)

We will encounter these functions more extensively in the superstring. The theta function satisﬁes Jacobi’s ‘abstruse identity,’ a special case of a quartic identity of Riemann, ϑ400 (0, τ) − ϑ401 (0, τ) − ϑ410 (0, τ) = 0 .

(7.2.41)

ϑ11 (0, τ) = 0 .

(7.2.42)

Note also that

216

7 One-loop amplitudes

Finally, the Dedekind eta function is η(τ) = q

1/24

∞

∂ν ϑ11 (0, τ) (1 − q ) = −2π m=1 m

1/3

(7.2.43)

.

It has the modular transformations η(τ + 1) = exp(iπ/12) η(τ) , η(−1/τ) = (−iτ) 7.3

1/2

(7.2.44a) (7.2.44b)

η(τ) .

The torus amplitude

We now apply to the torus the general result (5.3.9) expressing the string amplitude as an integral with ghost and vertex operator insertions. The two CKVs require one vertex operator to be ﬁxed, so ST 2 (1; 2; . . . ; n) 1 = 2

dτd¯τ F0

˜ ˜ccV1 (w1 , w ¯ 1) BB

n i=2

¯ i Vi (wi , w ¯ i) dwi dw

.

(7.3.1)

T2

The fundamental region F0 and the factor of 12 from w → −w were discussed in section 5.1. As usual for complex variables, dτd¯τ = 2dτ1 dτ2 , where τ = τ1 + iτ2 . The ghost insertion for dτ is 1 1 d2 w bww (w)∂τ gw¯ w¯ B= (b, ∂τ g) = 4π 2π i = d2 w bww (w) 4πτ2 → 2πibww (0) . (7.3.2) We have used the result (7.2.13) for the τ-derivative of the metric. In the ﬁnal line we have used the fact that the ghost path integral (7.2.27) is independent of position to carry out the w-integration. (Recall that d2 w = 2dσ 1 dσ 2 and that gww¯ = 12 .) The CKG consists of the translations of the torus. Since this group has ﬁnite volume, we need not ﬁx it: we can rewrite the amplitude so that all vertex operators are integrated and the volume of the CKG is divided out. It is easy to do this by hand here. The CKVs are constant, so the expectation value in eq. (7.3.1) is independent of where the c ghosts are placed: we can move them away from w1 , putting them at some ﬁxed position. For the vertex operators, translation invariance implies that the amplitude is unchanged if we replace wi → wi + w. Average over translations ¯ dwdw , (7.3.3) 2 2(2π) τ2

217

7.3 The torus amplitude

the denominator being the area of the torus, which is the volume of the CKG. Using eqs. (7.3.2) and (7.3.3), the amplitude becomes ST 2 (1; 2; . . . ; n)

= F0

dτd¯τ 4τ2

b(0)˜b(0)˜c(0)c(0)

n

¯ i Vi (wi , w ¯ i) dwi dw

i=1

.

(7.3.4)

T2

All vertex operators are now on an equal footing. The form (7.3.4) could also have been derived directly, without the intermediate step of ﬁxing a vertex operator, and is valid even without vertex operators,

ZT 2 =

F0

dτd¯τ b(0)˜b(0)˜c(0)c(0) 2 . T 4τ2

(7.3.5)

This vacuum amplitude is quite interesting and will be our main focus. Not only does it reveal the essential diﬀerence between the ultraviolet behaviors of string theory and ﬁeld theory, but it has an important physical interpretation of its own. For 26 ﬂat dimensions, the matter and ghost path integrals (7.3.5) were evaluated in the previous section, with the result

ZT 2 = iV26

F0

dτd¯τ (4π 2 α τ2 )−13 |η(τ)|−48 . 4τ2

(7.3.6)

The amplitude has the important property of modular invariance. The product τ2 |η(τ)|4 is invariant by eq. (7.2.44), and it is easy to check that dτd¯τ τ22

(7.3.7)

is modular-invariant as well. Note the exponent −48, from the 24 leftand right-moving oscillators. The contribution of the ghosts cancels two sets of oscillators, leaving only the contribution of the transverse modes. The amplitude with n tachyons is given by including the vertex operator expectation value (7.2.4) and integrating over positions. For a general CFT the path integral on the torus is expressed as a trace in eq. (7.2.28). As long as there are d ≥ 2 noncompact ﬂat dimensions the ghosts still cancel two sets of bosonic operators, and the vacuum amplitude is

ZT 2 = Vd

F0

= iVd

F0

dτd¯τ 4τ2

dd k ˜ exp(−πτ2 α k 2 ) q hi −1 q¯hi −1 d (2π) ⊥

(7.3.8a)

i∈H

dτd¯τ ˜ (4π 2 α τ2 )−d/2 q hi −1 q¯hi −1 . 4τ2 ⊥ i∈H

(7.3.8b)

218

7 One-loop amplitudes

Here H⊥ is the closed string Hilbert space excluding the ghosts, the µ = 0, 1 oscillators, and the noncompact momenta. In order to understand the physics of these amplitudes it is useful to compare with the corresponding quantity in ﬁeld theory, the sum over all particle paths with the topology of a circle. This is given by

∞ dl dd k ZS1 (m ) = Vd exp[−(k 2 + m2 )l/2] (2π)d 0 2l ∞ dl = iVd (7.3.9) (2πl)−d/2 exp(−m2 l/2) . 0 2l This can be derived by gauge-ﬁxing the point-particle path integral, which was given as exercise 5.2. The result is intuitive: l is the modulus for the circle, 12 (k 2 + m2 ) is the world-line Hamiltonian, and the 2l in the denominator removes the overcounting from translation and reversal of the world-line coordinate. Let us now take this point-particle result, which is for a particle of given m2 , and sum over the actual spectrum of the string. As shown in chapter 4, the physical spectrum is in one-to-one correspondence with H⊥ , where the mass is related to the transverse weights by 2 (7.3.10) m2 = (h + ˜h − 2) α and with the constraint h = ˜h. It is useful to write the latter constraint in integral form, π

dθ (7.3.11) δh,˜h = exp i(h − ˜h)θ . −π 2π In writing this we are assuming that the diﬀerence h − ˜h is an integer. This is true in 26 ﬂat dimensions, and as we have discussed in the previous section it is necessary in general for modular invariance. Then ∞ dl π dθ ZS1 (m2i ) = iVd (2πl)−d/2 2l 2π 0 −π ⊥ 2

i∈H

×

= iVd where θ +

il/α

R

exp −(hi + ˜hi − 2)l/α + i(hi − ˜hi )θ

i∈H⊥

dτd¯τ ˜ (4π 2 α τ2 )−d/2 q hi −1 q¯hi −1 , 4τ2 ⊥

(7.3.12)

i∈H

= 2πτ. Here the region of integration R is

1 . (7.3.13) 2 Now let us interpret these results. The amplitude (7.3.9) for a single point particle diverges as l → 0. This is the usual UV divergence of quantum R:

τ2 > 0 ,

|τ1 | <

219

7.3 The torus amplitude

ﬁeld theory (note that l has units of spacetime length-squared). Summing this over the string spectrum as in eq. (7.3.12) only makes things worse, as all the states of the string contribute with the same sign. However, compare the sum (7.3.12) to the actual string amplitude (7.3.8). There is great similarity but also one remarkable diﬀerence. The integrands are identical but the regions of integration are diﬀerent: in the string it is the fundamental region 1 (7.3.14) , τ2 > 0 2 and in ﬁeld theory the larger region R. In the string amplitude, the UV divergent region is simply absent. We can also see this from the point of view of the momentum integral (7.3.8a). Since τ2 is bounded below, this integral (after contour rotation) is a convergent Gaussian. Another possible divergence comes from the limit τ2 → ∞, where the torus becomes very long. In this region, the string amplitude in 26 ﬂat dimensions has the expansion F0 :

iV26

∞ dτ2

2τ2

|τ| > 1 ,

|τ1 | <

(4π 2 α τ2 )−13 exp(4πτ2 ) + 242 + . . . ,

(7.3.15)

the asymptotic behavior being controlled by the lightest string states. The ﬁrst term in the series diverges, and the interpretation is clear from the ﬁeld theory point of view. The series is in order of increasing masssquared. The ﬁrst term is from the tachyon and diverges due to the positive exponential in the path sum (7.3.9). This divergence is therefore an artifact of a theory with a tachyon, and will not aﬄict more realistic string theories. Incidentally, this path sum can be deﬁned by analytic continuation from positive mass-squared, but there is still a pathology — the continued energy density is complex. This signiﬁes an instability, the tachyon ﬁeld rolling down its inverted potential. With a general CFT, eq. (7.3.12) becomes iVd

∞ dτ2

2τ2

(4π 2 α τ2 )−d/2

exp(−πα m2i τ2 ).

(7.3.16)

i

Because the tachyon is always present in bosonic string theory this again diverges; in the absence of tachyons it will converge. The torus illustrates a general principle, which holds for all string amplitudes: there is no UV region of moduli space that might give rise to high energy divergences. All limits are controlled by the lightest states, the long-distance physics. For the torus with vertex operators the τ-integration is still cut oﬀ as above, but there are more limits to consider as vertex operators approach one another. The same general principle applies to these as well, as we will develop further in chapter 9.

220

7 One-loop amplitudes

One might try to remove the UV divergence from ﬁeld theory by cutting oﬀ the l integral. Similarly, one might try to make an analogous modiﬁcation of string theory, for example replacing the usual fundamental region F0 with the region |τ1 | ≤ 12 , τ2 > 1 whose lower edge is straight. However, in either case this would spoil the consistency of the theory: unphysical negative norm states would no longer decouple. We have seen in the general discussion in section 5.4 that the coupling of a BRSTnull state is proportional to a total derivative on moduli space. For the fundamental region F0 , the apparent boundaries I and I and II and II respectively (shown in ﬁg. 5.2) are identiﬁed and the surface terms cancel. Modifying the region of integration introduces real boundaries. Surface terms on the modiﬁed moduli space will no longer cancel, so there is nonzero amplitude for null states and an inconsistent quantum theory. This is directly analogous to what happens in gravity if we try to make the theory ﬁnite by some brute force cutoﬀ on the short-distance physics: it is exceedingly diﬃcult to do this without spoiling the local spacetime symmetry and making the theory inconsistent. String theory manages in a subtle way to soften the short-distance behavior and eliminate the divergences without loss of spacetime gauge invariance. Physics of the vacuum amplitude Besides serving as a simple illustration of the behavior of string amplitudes, the one-loop vacuum amplitude has an interesting physical interpretation. In point-particle theory, ‘vacuum’ paths consist of any number n of disconnected circles. Including a factor of 1/n! for permutation symmetry and summing in n leads to

Z vac (m2 ) = exp ZS1 (m2 ) .

(7.3.17)

Translating into canonical ﬁeld theory, Z vac (m2 ) = 0| exp(−iHT )|0

= exp(−iρ0 Vd ) , where ρ0 is the vacuum energy density: i ZS (m2 ) . ρ0 = Vd 1

(7.3.18)

(7.3.19)

The l integral in ZS1 (m2 ) diverges as l → 0. In renormalizable ﬁeld theory this divergence is canceled by counterterms or supersymmetry. To get a rough insight into the physics we will cut oﬀ the integral at l ≥ and then take → 0, but drop divergent terms. With this prescription, ∞ dl 0

1 exp[−(k 2 + m2 )l/2] → − ln(k 2 + m2 ) 2l 2

(7.3.20)

221

7.3 The torus amplitude and i

∞ dl ∞ dk 0 0

2l

−∞

2π

exp[−(k 2 + m2 )l/2] →

ωk , 2

(7.3.21)

where ωk2 = k · k + m2 . Using the latter form, the vacuum energy density becomes

ρ0 =

dd−1 k ωk , (2π)d−1 2

(7.3.22)

which is just the sum of the zero-point energies of the modes of the ﬁeld. This is in spacetime, of course — we encountered the same kind of sum but on the world-sheet in section 1.3. The description of a quantum ﬁeld theory as a sum over particle paths is not as familiar as the description in terms of a sum over ﬁeld histories, but it is equivalent. In particular, the free path integral in ﬁeld theory can be found in most modern ﬁeld theory textbooks: 1 ln Z vac (m2 ) = − Tr ln(−∂2 + m2 ) 2 Vd dd k =− ln(k 2 + m2 ). 2 (2π)d

(7.3.23)

Using eq. (7.3.20), this is the same as the path sum result (7.3.9). In older quantum ﬁeld theory texts, one often reads that vacuum amplitudes such as eq. (7.3.18) are irrelevant because they give an overall phase that divides out of any expectation value. This is true if one is considering scattering experiments in a ﬁxed background and ignoring gravity, but there are at least two circumstances in which the vacuum energy density is quite important. The ﬁrst is in comparing the energy densities of diﬀerent states to determine which is the true ground state of a theory. It is very likely, for example, that the breaking of the electroweak S U(2)×U(1) symmetry is determined in part by such quantum corrections to the vacuum energy density. This was the original motivation that led to the Coleman–Weinberg formula (7.3.23). The generalization to a theory with particles of arbitrary spin is ρ0 =

i (−1)Fi ZS1 (m2i ). Vd i

(7.3.24)

The sum runs over all physical particle states. Each polarization counts separately, giving a factor of 2si + 1 for a particle of spin si in four dimensions. Here Fi is the spacetime fermion number, deﬁned mod 2, so fermions contribute with the opposite sign. The second circumstance is when one considers the coupling to gravity. The vacuum energy gives a source term in Einstein’s equation, the

222

7 One-loop amplitudes

cosmological constant, and so has observable eﬀects. In fact, this cosmological constant is a great challenge, because the fact that spacetime is approximately ﬂat and static means that its value is very small, −44 |ρ0 | < GeV 4 . ∼ 10

(7.3.25)

If one considers only the contributions to the vacuum energy from the vacuum ﬂuctuations of the known particles up to the currently explored energy (roughly the electroweak scale mew ), the zero-point energy is already of order m4ew ≈ 108 GeV 4 ,

(7.3.26)

52 orders of magnitude too large. The Higgs ﬁeld potential energy and the QCD vacuum energy are also far too large. Finding a mechanism that would account for the cancellation of the net cosmological constant to great accuracy has proven very diﬃcult. For example, in a supersymmetric theory, the contributions of degenerate fermions and bosons in the sum (7.3.24) cancel. But supersymmetry is not seen in nature, so it must be a broken symmetry. The cancellation is then imperfect, leaving again a remainder of at least m4ew . The cosmological constant problem is one of the most nagging diﬃculties, and therefore probably one of the best clues, in trying to ﬁnd a uniﬁed theory with gravity. What about the cosmological constant problem in string theory? At string tree level we had a consistent theory with a ﬂat metric, so the cosmological constant was zero. In fact we arranged this by hand, when we took 26 dimensions. One sees from the spacetime action (3.7.20) that there would otherwise be a tree-level potential energy proportional to D − 26. The one-loop vacuum energy density in bosonic string theory is nonzero, and is necessarily of the order of the string scale (ignoring the tachyon divergence). In four dimensions this would correspond to 1072 GeV 4 , which is again far too large. In a supersymmetric string theory, there will be a certain amount of cancellation, but again one expects a remainder of at least m4ew in a realistic theory with supersymmetry breaking. The cosmological constant problem is telling us that there is something that we still do not understand about the vacuum, in ﬁeld theory and string theory equally. We will return to this problem at various points. 7.4

Open and unoriented one-loop graphs The cylinder

The results for the torus are readily extended to the other surfaces of Euler number zero. For example, the vacuum amplitude from the cylinder

7.4 Open and unoriented one-loop graphs

223

σ2 σ1 (a)

(b)

(c) Fig. 7.1. (a) Cylinder in the limit of small t. (b) The amplitude separated into disk tadpole amplitudes and a closed string propagator. (c) Analogous ﬁeld theory graph. The heavy circles represent the tadpoles.

in the oriented theory is ZC2 =

∞ dt 0

= iVd

2t

Tro [exp(−2πtL0 )]

∞ dt 0

2t

(8π 2 α t)−d/2

∞ dt

exp[−2πt(hi − 1)]

i∈H⊥ o

(7.4.1) (8π 2 α t)−13 η(it)−24 . 2t 0 This can be obtained either by working out the path integral measure in terms of the ghost zero modes, as we did for the torus, or by guessing (correctly) that we should again sum the point-particle result (7.3.9) over the open string spectrum. In the ﬁrst line, the trace is over the full open string CFT, except that for omission of the ghost zero modes as denoted by the prime. In the ﬁnal line the trace has been carried out for the case of 26 ﬂat dimensions, with n Chan–Paton degrees of freedom. Including tachyon vertex operators is straightforward. The t → ∞ limit of the cylinder is much like the τ2 → ∞ limit of the torus. The cylinder looks like a long strip, and the leading asymptotics are given by the lightest open string states. As in the closed string there is a divergence, but only from the open string tachyon. The t → 0 limit is rather interesting. Unlike the case of the torus, there is no modular group acting to cut oﬀ the range of integration, and so the UV divergence of ﬁeld theory still seems to be present. However, we will see that, as with all divergences in string theory, this should actually be interpreted as a long-distance eﬀect. In the t → 0 limit the cylinder is very long as shown in ﬁgure 7.1(a). That is, it looks like a closed string appearing from the vacuum, propagating a distance, and then disappearing again into the vacuum. To make this clear, use the → iV26 n

2

224

7 One-loop amplitudes

modular transformation (7.2.44) η(it) = t−1/2 η(i/t)

(7.4.2)

and change variables to s = π/t, with the result ZC2 = i

V26 n2 2π(8π 2 α )13

∞

ds η(is/π)−24 .

(7.4.3)

0

Scaling down the metric by 1/t so that the cylinder has the usual closed string circumference 2π, the length of the cylinder is s. Expanding η(is/π)−24 = exp(2s)

∞

[1 − exp(−2ns)]−24

n=1

= exp(2s) + 24 + O(exp(−2s)) ,

(7.4.4)

one sees the expected asymptotics from expanding in a complete set of closed string states. In other words, if we think of σ 2 as world-sheet time and σ 1 as world-sheet space, ﬁgure 7.1(a) is a very short open string loop. If we reverse the roles of world-sheet space and time, it is a very long closed string world-line beginning and ending on boundary loops. In the Euclidean path integral, either description can be used, and each is useful in a diﬀerent limit of moduli space. The leading divergence in the vacuum amplitude is from the closed string tachyon. This is uninteresting and can be deﬁned by analytic continuation, ∞ 0

ds exp(βs) ≡ −

1 . β

(7.4.5)

The second term, from the massless closed string states, gives a divergence of the form 1/0 even with the continuation. To see the origin of this divergence, imagine separating the process as shown in ﬁgure 7.1(b). As we have discussed in section 6.6, there is a nonzero amplitude with one closed string vertex operator on the disk. This tadpole corresponds to a closed string appearing from or disappearing into the vacuum. In between the two tadpoles is the closed string propagator. In momentum space the massless propagator is proportional to 1/k 2 . Here, momentum conservation requires that the closed string appear from the vacuum with zero momentum, so the propagator diverges. This same kind of divergence occurs in quantum ﬁeld theory. Consider a massless scalar ﬁeld φ with a term linear in φ in the Lagrangian. There is then a vertex which connects to just a single propagator, and the graph shown in ﬁgure 7.1(c) exists. This diverges because the intermediate propagator is

1 . k 2 kµ =0

(7.4.6)

7.4 Open and unoriented one-loop graphs

225

This is a long-distance (IR) divergence, because poles in propagators are associated with propagation over long spacetime distances. UV and IR divergences in quantum ﬁeld theory have very diﬀerent physical origins. UV divergences usually signify a breakdown of the theory, the need for new physics at some short distance. IR divergences generally mean that we have asked the wrong question, or expanded in the wrong way. So it is here. In normal perturbation theory we expand around φ(x) = 0, or some other constant conﬁguration. For the action 1 − 2 g

d

d x

1 ∂µ φ∂µ φ + gΛφ 2

,

(7.4.7)

the equation of motion ∂2 φ = gΛ

(7.4.8)

does not allow φ(x) = 0 as a solution. Instead we must expand around a solution to the equation (7.4.8); any solution is necessarily positiondependent. The resulting amplitudes are free of divergences. This is true even though the right-hand side is a perturbation (we have included the factor of g appropriate to the disk), because it is a singular perturbation. In particular, the corrected background breaks some of the Poincar´e symmetry of the zeroth order solution. The situation is just the same in string theory. The disk tadpole is a source −Λ

˜

d26 x (−G)1/2 e−Φ

(7.4.9)

for both the dilaton and metric. Expanding around a solution to the corrected ﬁeld equations (with the metric and dilaton no longer constant) leads to ﬁnite amplitudes. The details are a bit intricate, and will be discussed somewhat further in chapter 9. Incidentally, in supersymmetric string theories if the tree-level background is invariant under supersymmetry, it usually receives no loop corrections. The pole (7.4.6) is the same kind of divergence encountered in the tree-level amplitudes, a resonance corresponding to propagation over long spacetime distances. If we add open string vertex operators to each end of the cylinder so that it represents a one-loop open string scattering amplitude, then the momentum k µ ﬂowing from one boundary to the other is in general nonzero. The large-s limit (7.4.4) then includes a factor of exp(−α k 2 s/2)

(7.4.10)

and the divergence becomes a momentum pole representing scattering of open strings into an intermediate closed string. Thus, as claimed in chapter 3, an open string theory must include closed strings as well. It is

226

7 One-loop amplitudes

curious that the mechanism that removes the UV divergence is diﬀerent in the closed and open string cases. In the closed string it is an eﬀective cutoﬀ on the modular integration; in the open string it is a reinterpretation of the dangerous limit of moduli space in terms of long distances. In eq. (7.4.1) the path integral on the cylinder has been related to a trace over the open string spectrum by cutting open the path integral with σ 2 treated as time. It can also be obtained in terms of the closed string Hilbert space by treating σ 1 as time. Let σ 2 be periodic with period 2π and the boundaries be at σ 1 = 0, s. The closed string appears in some state |B at σ 1 = 0 and then disappears in the same way at σ 1 = s. Including the measure insertions, the path integral is then proportional to ˜ 0 )]|B .

B|c0 b0 exp[−s(L0 + L (7.4.11) The boundary state |B is determined by the condition that ∂1 X µ , c1 , and b12 vanish on the boundary. In the Hamiltonian form these must annihilate |B , which in terms of the Laurent coeﬃcients is (αµ + ˜αµ−n )|B = (cn + ˜c−n )|B = (bn − ˜b−n )|B = 0 , all n . (7.4.12) n

This determines

|B ∝ (c0 + ˜c0 ) exp −

∞

−1

(n α−n · ˜α−n + b−n˜c−n + ˜b−n c−n ) |0; 0 . (7.4.13)

n=1

Using this in (7.4.11) gives the result (7.4.3), except that the normalization of |B is undetermined. This representation is useful in analyzing the t → 0 limit and the closed string poles. By comparing the string and ﬁeld theory calculations we can determine the disk tadpole Λ, but it will be more convenient to do this in the next chapter as a special case of a more general result. The Klein bottle The vacuum amplitude from the Klein bottle is ZK 2 =

∞ dt 0

= iVd

4t

#

˜ 0 )] Trc Ω exp[−2πt(L0 + L

∞ dt 0

4t

(4π 2 α t)−d/2

$

Ωi exp[−2πt(hi + ˜hi − 2)] ,

(7.4.14)

i∈H⊥ c

where the notation follows the cylinder (7.4.1). Relative to the torus in the oriented theory there is an extra factor of 12 from the projection operator 1 1 2 (1 + Ω). For the same reason there is an extra 2 in both the torus and cylinder amplitudes in the unoriented theory. One can also think of this ¯ To evaluate the trace as coming from the extra gauge invariance w → w. for 26 ﬂat dimensions, note that the only diagonal elements of Ω are those

7.4 Open and unoriented one-loop graphs

227

σ2 σ1 (a)

(b)

Fig. 7.2. (a) Klein bottle in the limit of small t as a cylinder capped by crosscaps. (b) The amplitude separated into RP2 tadpole amplitudes and a closed string propagator.

for which the right- and left-movers are in the same state, and these states contribute with Ω = +1. The trace is then eﬀectively over only one side, which is the same as the open string spectrum, except that the weight is doubled because the right- and left-movers make equal contributions. The result is ∞ dt (4π 2 α t)−13 η(2it)−24 . (7.4.15) ZK2 → iV26 4t 0 The modulus t runs over the same range 0 < t < ∞ as for the cylinder, so the t → 0 divergences are the same. Again the 1/0 pole has a longdistance interpretation in terms of a closed string pole. To see this consider the regions 0 ≤ σ 1 ≤ 2π , 0 ≤ σ 2 ≤ 2πt , 0 ≤ σ 1 ≤ π , 0 ≤ σ 2 ≤ 4πt , each of which is a fundamental region for the identiﬁcation ¯ + 2πit . w∼ = w + 2π ∼ = −w

(7.4.16a) (7.4.16b) (7.4.17)

In (7.4.16a), the left- and right-hand edges are periodically identiﬁed while the upper and lower edges are identiﬁed after a parity-reversal, giving an interpretation as a closed string loop weighted by Ω. For (7.4.16b), note that the identiﬁcations (7.4.17) imply also that ¯ + π) + 2πit . w∼ (7.4.18) = w + 4πit , w + π ∼ = −(w It follows that the upper and lower edges of the region (7.4.16b) are periodically identiﬁed. The left-hand edge is identiﬁed with itself after translation by half its length and reﬂection through the edge, and similarly the right-hand edge: this is the deﬁnition of a cross-cap. Thus we have the representation in ﬁgure 7.2(a) as a cylinder capped by two cross-caps. Rescaling by 1/2t, the cylinder has circumference 2π and length s = π/2t. After a modular transformation, the amplitude becomes ZK2

226 V26 =i 4π(8π 2 α )13

∞ 0

ds η(is/π)−24 .

(7.4.19)

228

7 One-loop amplitudes

σ2 σ1 (a)

(b)

Fig. 7.3. (a) M¨ obius in the limit of small t as a cylinder capped by one cross-cap. (b) The amplitude separated into D2 and RP2 tadpole amplitudes and a closed string propagator.

All the discussion of the cylinder divergence now applies, except that the tadpole is from the projective plane rather than the disk. The M¨obius strip For the M¨ obius strip, ZM2 = iVd

∞ dt 0

4t

(8π 2 α t)−d/2

Ωi exp[−2πt(hi − 1)]

(7.4.20)

i∈H⊥ o

This diﬀers from the result on the cylinder only by the Ω and the factor of 12 from the projection operator. In 26 ﬂat dimensions, the eﬀect of the operator Ω in the trace is an extra −1 at the even mass levels plus an appropriate accounting of the Chan–Paton factors. The oscillator trace is thus exp(2πt)

∞

[1 − (−1)n exp(−2πnt)]−24 = ϑ00 (0, 2it)−12 η(2it)−12 . (7.4.21)

n=1

For the S O(n) theory the 12 n(n + 1) symmetric states have Ω = +1 while the 12 n(n − 1) antisymmetric states have Ω = −1 for a net contribution of n. For the S p(k) theory these degeneracies are reversed, giving −n (recall that in our notation S p(k) corresponds to n = 2k Chan–Paton states). The amplitude is then ZM2 = ±inV26

∞ dt 0

4t

(8π 2 α t)−13 ϑ00 (0, 2it)−12 η(2it)−12 .

(7.4.22)

By the same construction as for the Klein bottle, the M¨ obius strip can be represented as a cylinder with a boundary at one end and a cross-cap on the other, as in ﬁgure 7.3(a). The length of the cylinder is now s = π/4t. By a modular transformation, the amplitude becomes ZM 2

213 V26 = ±2in 4π(8π 2 α )13

∞ 0

ds ϑ00 (0, 2is/π)−12 η(2is/π)−12 .

(7.4.23)

Exercises

229

As for the annulus this can be written as an operator expression, one of the boundary states in eq. (7.4.11) being replaced by an analogous cross-cap state |C . There is again a t → 0 divergence; it corresponds to the process of ﬁgure 7.3(a) with one tadpole from the disk and one from the projective plane. In the unoriented theory the divergences on these three surfaces combine into ∞ 24V26 13 2 (2 ∓ n) ds . (7.4.24) i 4π(8π 2 α )13 0 That is, the total tadpole is proportional to 213 ∓ n. For the gauge group S O(213 ) = S O(8192) this vanishes, the tadpoles from the disk and projective plane canceling. For the bosonic string this probably has no special signiﬁcance, but the analogous cancellation for S O(32) in superstring theory does. Exercises 7.1 Fill in the steps leading to the expectation value (7.2.4) of exponential operators on the torus. Show that it has the correct transformation under τ → −1/τ. 7.2 Derive the torus vacuum amplitude (7.3.6) by regulating and evaluating the determinants, as is done for the harmonic operator in appendix A. Show that a modular transformation just permutes the eigenvalues. [Compare exercise A.3.] (Reference: Polchinski (1986).) 7.3 Derive the Green’s function (7.2.3) by carrying out the eigenfunction sum (6.2.7). 7.4 Evaluate ∂w X µ (w)∂w Xµ (0) on the torus by representing it as a trace. Show that the result agrees with eq. (7.2.16). 7.5 (a) Obtain the leading behavior of the theta functions (7.2.37) as Im τ → ∞. (b) By using the modular transformations, do the same as τ → 0 along the imaginary axis. (c) A more exotic question: what if τ approaches a nonzero point on the real axis, along a path parallel to the imaginary axis? [Hint: the answer depends crucially on whether the point is rational or irrational.] 7.6 (a) Verify eqs. (7.3.20) and (7.3.21). (b) Evaluate ZS1 (m2 ) with a cutoﬀ on the l-integral in any dimension d. Show that the counterterms are analytic in m2 . Give the nonanalytic part for d = 4. (Reference: Coleman & Weinberg (1973).) 7.7 Obtain the amplitude for n open string tachyons on the cylinder

230

7 One-loop amplitudes

in a form analogous to the torus amplitude (eq. (7.3.6) combined with eq. (7.2.4)). The necessary Green’s function can be obtained from that on the torus by the method of images, using the representation of the cylinder as an involution of the torus. 7.8 Consider the amplitude from the previous problem, in the case that there are some vertex operators on each of the two boundaries. Show from the explicit result of the previous problem that the t → 0 limit gives a series of closed string poles as a function of the momentum ﬂowing from one boundary to the other. 7.9 If you have done both exercises 6.12 and 7.8, argue that unitarity relates the square of the former to the tachyon pole of the latter. Use this to ﬁnd the numerical relation between gc and go2 . We will do something similar to this, in a more roundabout way, in the next chapter. 7.10 Argue that unitarity of the cylinder amplitude with respect to intermediate open strings requires the Chan–Paton factors to satisfy Tr (λa λb )Tr (λc λd ) = Tr (λa λb λe λc λd λe ). Show that the gauge group U(n1 ) × U(n2 ) × . . . × U(nk ) is consistent with unitarity at tree level, but that this additional condition singles out U(n). 7.11 This exercise reproduces the ﬁrst appearance of D = 26 in string theory (Lovelace, 1971). Calculate the cylinder vacuum amplitude from the Coleman–Weinberg formula in the form (7.3.9), assuming D spacetime dimensions and D net sets of oscillators. Fix the constant in the open string mass-squared so that the tachyon remains at m2 = −1/α , as required by conformal invariance of the vertex operator. Insert an additional factor of exp(−α k 2 s/2) as in eq. (7.4.10) to simulate a scattering amplitude. Show that unless D = 26 and D = 24, the amplitude has branch cuts (rather than poles) in k 2 . 7.12 Repeat the previous exercise for the torus, showing in this case that the vacuum amplitude is modular-invariant only for D = 26 and D = 24 (Shapiro, 1972). 7.13 Carry out the steps, parallel to those used to derive the torus amplitude (7.3.6), to obtain the vacuum Klein bottle amplitude (7.4.15) from the string path integral. 7.14 Show that the amplitude (7.4.11) with the boundary state (7.4.13) reproduces the cylinder amplitude (7.4.3). 7.15 (a) Find the state |C corresponding to a cross-cap. (b) By formulae analogous to eq. (7.4.11), obtain the Klein bottle and M¨ obius strip vacuum amplitudes.

8 Toroidal compactiﬁcation and T -duality

Realistic compactiﬁcations of string theory will be the subject of the later chapters of volume two, but it is interesting to look now at the simplest compactiﬁcation of string theory, in which one or more dimensions are periodically identiﬁed. We ﬁrst consider the same compactiﬁcation in ﬁeld theory, encountering in particular the Kaluza–Klein uniﬁcation of gauge interactions and gravity. We then extend this to string theory, where several new and intrinsically stringy phenomena arise: winding states, enhanced gauge symmetries, T -duality, and D-branes. We are also led to consider slightly more complicated compactiﬁcations, namely orbifolds and orientifolds. 8.1

Toroidal compactiﬁcation in ﬁeld theory

In general relativity, the geometry of spacetime is dynamical. The three spatial dimensions we see are expanding and were once highly curved. It is a logical possibility that there are additional dimensions that remain small. In fact, this was put forward as early as 1914 as a means of unifying the electromagnetic and gravitational ﬁelds as components of a single higherdimensional ﬁeld. Consider the case of a ﬁve-dimensional theory, with x4 periodic, (8.1.1) x4 ∼ = x4 + 2πR , and with xµ noncompact for µ = 0, . . . , 3. This is toroidal compactiﬁcation. The ﬁve-dimensional metric separates into Gµν , Gµ4 , and G44 . From the four-dimensional point of view, these are a metric, a vector, and a scalar. Let us see this in detail, taking the more general case of D = d + 1 spacetime dimensions with xd periodic. Since we are still in ﬁeld theory we leave D arbitrary. Parameterize the metric as M N µ ν d µ 2 ds2 = GD MN dx dx = Gµν dx dx + Gdd (dx + Aµ dx ) .

231

(8.1.2)

232

8 Toroidal compactiﬁcation and T -duality

Indices M, N run over all dimensions 0, . . . , d and indices µ, ν run only over noncompact dimensions 0, . . . , d − 1. We designate the full D-dimensional D metric by GD MN . One should note that Gµν = Gµν ; in d-dimensional actions we raise and lower indices with Gµν . For now the ﬁelds Gµν , Gdd , and Aµ are allowed to depend only on the noncompact coordinates xµ . The form (8.1.2) is the most general metric invariant under translations of xd . This form still allows d-dimensional reparameterizations xµ (xν ) and also reparameterizations xd = xd + λ(xµ ) .

(8.1.3)

Aµ = Aµ − ∂µ λ ,

(8.1.4)

Under the latter,

so gauge transformations arise as part of the higher-dimensional coordinate group. This is the Kaluza–Klein mechanism. To see the eﬀect of xd -dependence, consider a massless scalar φ in D dimensions, where for simplicity the metric in the compact direction is Gdd = 1. The momentum in the periodic dimension is quantized, pd = n/R. Expand the xd -dependence of φ in a complete set, φ(xM ) =

∞

φn (xµ ) exp(inxd /R) .

(8.1.5)

n=−∞

The D-dimensional wave equation ∂M ∂M φ = 0 becomes n2 φn (xµ ) . (8.1.6) R2 The modes φn of the D-dimensional ﬁeld thus become an inﬁnite tower of d-dimensional ﬁelds, labeled by n. The d-dimensional mass-squared ∂µ ∂µ φn (xµ ) =

n2 (8.1.7) R2 is nonzero for all ﬁelds with nonvanishing pd . At energies small compared to R −1 , only the xd -independent ﬁelds remain and the physics is d-dimensional. At energies above R −1 , one sees the tower of Kaluza–Klein states. The charge corresponding to the Kaluza–Klein gauge invariance (8.1.3) is the pd -momentum. In this simple example, all ﬁelds carrying the Kaluza– Klein charge are massive. More generally, with higher spin ﬁelds and curved backgrounds, there can be massless charged ﬁelds. The eﬀective action for the massless ﬁelds is always an important object to consider. Deﬁne Gdd = e2σ . The Ricci scalar for the metric (8.1.2) is − p µ pµ =

1 R = Rd − 2e−σ ∇2 eσ − e2σ Fµν F µν , 4

(8.1.8)

8.1 Toroidal compactiﬁcation in ﬁeld theory

233

where R is constructed from GD MN and Rd from Gµν . The graviton-dilaton action (3.7.20) becomes

1 dD x (−GD )1/2 e−2Φ (R + 4∇µ Φ∇µ Φ) S1 = 2κ20 πR = 2 dd x (−Gd )1/2 e−2Φ+σ κ0 1 2σ µ µ µν × Rd − 4∂µ Φ∂ σ + 4∂µ Φ∂ Φ − e Fµν F 4 πR = 2 dd x (−Gd )1/2 e−2Φd κ0 1 × Rd − ∂µ σ∂µ σ + 4∂µ Φd ∂µ Φd − e2σ Fµν F µν , 4

(8.1.9)

giving kinetic terms for all the massless ﬁelds. Here Gd denotes the determinant of Gµν , and in the last line we have introduced the eﬀective d-dimensional dilaton, Φd = Φ − σ/2. The apparent wrong sign of the dilaton kinetic term is illusory because mixing of the graviton with the trace of the metric must also be taken into account. This is most easily done by means of a Weyl transformation as in eq. (3.7.25), and the resulting kinetic term has the correct sign. The ﬁeld equations do not determine the radius of the compact dimension:1 a ﬂat metric and constant dilaton are a solution for any values of Φ and σ. In other words, there is no potential energy for Φ or for σ, and so these ﬁelds are massless, much like Goldstone bosons. The diﬀerent values of Φ and σ label degenerate conﬁgurations (or states in the quantum theory), and a state in which these ﬁelds are slowly varying has energy only from the gradient. The diﬀerence from the Goldstone phenomenon is that the degenerate states are not related to one another by any symmetry, and in fact the physics depends on Φ and σ. In this bosonic theory the degeneracy is accidental and the one-loop energy, discussed in the previous chapter, breaks the degeneracy. In supersymmetric theories, the existence of physically inequivalent but degenerate vacua is quite common and plays an important role in understanding the dynamics. The massless ﬁelds, which label the inequivalent vacua, are called moduli. In nature, 1

Of course, only the invariant radius ρ = Reσ distinguishes physically inequivalent solutions. We could set R to some convenient value, say α1/2 , or to unity if we use dimensionless variables, but it is convenient to leave it general; often it will be more convenient to set Gdd to unity instead. Similarly we could set κ20 to be the appropriate power of α by an additive shift of Φ, but again it does not hurt to leave it general and we will see in chapter 13 that it is natural to ﬁx the additive normalization of Φ in a diﬀerent way.

234

8 Toroidal compactiﬁcation and T -duality

supersymmetry breaking must almost surely give mass to all moduli, else they would mediate inﬁnite-range interactions of roughly gravitational strength. We will see in chapter 14 that in some string theories the dilaton itself is the radius of a hidden dimension, one that is not evident in string perturbation theory. ˜ µ , the covariant derivative is Deﬁning Aµ = R A ˜µ , ∂µ + ipd Aµ = ∂µ + inA

(8.1.10)

so that the charges are integers. The d-dimensional gauge and gravitational ˜ µν ˜µν F couplings are conventionally deﬁned as follows. The coeﬃcient of F 2 in the Lagrangian density is deﬁned to be −1/4gd , and that of Rd is deﬁned to be 1/2κ2d . The gauge coupling is thus determined in terms of the gravitational coupling, gd2 =

κ20 e2Φd 2κ2d = . πR 3 e2σ ρ2

(8.1.11)

The d-dimensional and D-dimensional gravitational couplings are related 2πρ 1 = 2 , 2 κ κd

(8.1.12)

2πρ being the volume of the compactiﬁed dimension. The antisymmetric tensor also gives rise to a gauge symmetry by a generalization of the Kaluza–Klein mechanism. Separating BMN into Bµν and Aµ = Bdµ , the gauge parameter ζM deﬁned in eq. (3.7.7) separates into a d-dimensional antisymmetric tensor transformation ζµ and an ordinary gauge invariance ζd . The gauge ﬁeld is Bdµ and the ﬁeld strength Hdµν . The antisymmetric tensor action becomes

1 dD x (−GD )1/2 e−2Φ HMNL H MNL 24κ20 πR d 1/2 −2Φd ˜ ˜ µνλ + 3e−2σ H H µν . =− d x (−G ) e H H d µνλ dµν d 12κ20 (8.1.13)

S2 = −

We have deﬁned ˜ µνλ = (∂µ Bνλ − Aµ Hdνλ ) + cyclic permutations . H

(8.1.14)

The term proportional to the vector potential arises from the inverse metric GMN . It is known as a Chern–Simons term, this signifying the antisymmetrized combination of one gauge potential and any number of ﬁeld strengths. Such terms appear in many places in supersymmetric theories, and are associated with interesting physical eﬀects. Notice in

8.2 Toroidal compactiﬁcation in CFT

235

˜ µνλ is gauge-invariant because the variation (8.1.4) of Aµ particular that H is canceled by a variation Bνλ = Bνλ − λHdνλ .

(8.1.15)

There is no way to couple the potential BMN minimally to other ﬁelds, and so unlike the Kaluza–Klein case there are no ﬁelds charged under the antisymmetric tensor gauge symmetry; this will be diﬀerent in string theory.

8.2

Toroidal compactiﬁcation in CFT

Now we consider the conformal ﬁeld theory of a single periodic scalar ﬁeld, X∼ = X + 2πR .

(8.2.1)

To keep the equations uncluttered we drop the superscript from X d and world-sheet action is as in the noncompact theory, set 2 Gdd = 1. The , so the equations of motion, operator products, and ¯ d z ∂X ∂X/2πα energy-momentum tensor are unchanged. In particular, the theory is still conformally invariant. The periodicity has two eﬀects. First, string states must be single-valued under the identiﬁcation (8.2.1). That is, the operator exp(2πiRp) which translates strings once around the periodic dimension must leave states invariant, so the center-of-mass momentum is quantized k=

n , R

n∈Z.

(8.2.2)

This is just as in ﬁeld theory. The second eﬀect is special to string theory. A closed string may now wind around the compact direction, X(σ + 2π) = X(σ) + 2πRw ,

w∈Z.

(8.2.3)

The integer w is the winding number. States of winding numbers +1, 0, and −1 are shown in ﬁgure 8.1(a). From the point of view of the world-sheet ﬁeld theory, strings of nonzero winding number are topological solitons, states with a topologically nontrivial ﬁeld conﬁguration. A consistent string theory must include the winding number states: by the usual splitting–joining process a w = 0 string can turn into a w = +1, w = −1 pair as shown in ﬁgure 8.1(b). It is easy to see that winding number is always conserved as in this example. To describe the states of the closed string CFT, consider the Laurent

236

8 Toroidal compactiﬁcation and T -duality

X _1

0

+1

Y (a)

X 0

Y

_1

+1

(b)

Fig. 8.1. (a) Closed oriented strings of winding number w = +1, 0, −1. For illustration, one compact dimension X and one noncompact dimension Y are shown. (b) Transition of a w = 0 string into w = +1 and w = −1 strings.

expansions ∂X(z) = −i

1/2 ∞ α

αm

m+1 m=−∞ z

2

,

¯ z ) = −i α ∂X(¯ 2

1/2 ∞

˜αm . (8.2.4) m+1 ¯ z m=−∞

The total change in the coordinate X in going around the string is

2πRw =

¯ = 2π(α /2)1/2 (α0 − ˜α0 ) . (dz ∂X + d¯z ∂X)

(8.2.5)

The total Noether momentum is p=

1 2πα

¯ = (2α )−1/2 (α0 + ˜α0 ) . (dz ∂X − d¯z ∂X)

(8.2.6)

For a noncompact dimension this would give the usual relation α0 = ˜α0 = p(α /2)1/2 , but for the periodic dimension we have '

(1/2

n wR + , R α ' wR n (1/2 − . α˜0 = pR ≡ 2/α R α The Virasoro generators are pL ≡ 2/α

α0 =

(8.2.7a) (8.2.7b)

∞ α p2L L0 = α−n αn , + 4 n=1

(8.2.8a)

∞ 2 ˜ 0 = α pR + ˜α−n ˜αn . L 4 n=1

(8.2.8b)

237

8.2 Toroidal compactiﬁcation in CFT The partition function The partition function for X is now

˜

(q¯ q )−1/24 Tr q L0 q¯L0 = |η(τ)|−2 = |η(τ)|

∞

2

2

q α pL /4 q¯α pR /4

n,w=−∞ ∞ −2

exp −πτ2

n,w=−∞

α n2 w 2 R 2 + R2 α

+ 2πiτ1 nw

.

(8.2.9)

The oscillator sum is the same as in the noncompact case, while the momentum integration is replaced by a sum over n and w. Modular invariance is not manifest, but can be made so by using the Poisson resummation formula, ∞

2

exp(−πan + 2πibn) = a

n=−∞

∞

π(m − b)2 exp − . a m=−∞

−1/2

(8.2.10)

The partition function becomes 2πRZX (τ)

∞

exp −

m,w=−∞

πR 2 |m − wτ|2 α τ2

.

(8.2.11)

Here ZX (τ) is the modular-invariant expression (7.2.9) from the noncompact theory. The sum is obviously invariant under τ → τ + 1, by making a change of variables m → m + w. It is also invariant under τ → −1/τ with m → −w, w → m. The expression (8.2.11) has a simple path integral interpretation. Summing over all genus one world-sheets in a periodic spacetime, each nontrivial closed curve on the world-sheet can be wound around the compact direction, X(σ 1 + 2π, σ 2 ) = X(σ 1 , σ 2 ) + 2πwR , X(σ 1 + 2πτ1 , σ 2 + 2πτ2 ) = X(σ 1 , σ 2 ) + 2πmR .

(8.2.12a) (8.2.12b)

That is, the path integral breaks up into topologically distinct sectors labeled by w and m. The Gaussian path integration can be done by the usual strategy of writing X as a classical solution of the correct periodicity, Xcl = σ 1 wR + σ 2 (m − wτ1 )R/τ2 ,

(8.2.13)

plus a quantum piece with periodic boundary conditions. The action separates into a classical piece plus a quantum piece. The path integral over the quantum piece is just as in the noncompact case, while the classical action appears as the exponent in eq. (8.2.11). The eﬀect of the

238

8 Toroidal compactiﬁcation and T -duality

modular transformations on the periodicities is then simply to change the summation variables m and w. Vertex operators In order to create a winding state, which has α0 = ˜α0 , we need independent variables xL and xR , with [ xL , pL ] = [ xR , pR ] = i .

(8.2.14)

The ﬁeld X then splits into holomorphic and antiholomorphic parts, X(z, ¯z ) = XL (z) + XR (¯z ) , with

α α XL (z) = xL − i pL ln z + i 2 2

α α XR (¯z ) = xR − i pR ln ¯z + i 2 2

1/2 ∞ m=−∞ m=0

1/2 ∞

m=−∞ m=0

(8.2.15) αm , mz m

(8.2.16a)

˜αm . m¯z m

(8.2.16b)

If we restrict attention to states with kL = kR and to operators constructed only from the sum XL (z)+XR (¯z ), this reduces to the CFT for a noncompact dimension. Let us now discuss the OPE and vertex operators, ﬁrst in a slightly heuristic way and then ﬁlling in the details. It is easy to guess what the operator products should be, since the usual XX operator product −(α /2) ln(z12 ¯z12 ) separates into a sum of holomorphic plus antiholomorphic functions. Thus, α ln z12 , 2 XL (z1 )XR (¯z2 ) ∼ 0 .

XL (z1 )XL (z2 ) ∼ −

XR (¯z1 )XR (¯z2 ) ∼ −

α ln ¯z12 , 2

(8.2.17a) (8.2.17b)

The operator corresponding to the state |0; kL , kR would be VkL kR (z, ¯z ) = : eikL XL (z)+ikR XR (¯z ) :

(8.2.18)

with OPE α k kL /2 α kR kR /2 ¯z12 V(k+k )L (k+k )R (z2 , ¯z2 )

VkL kR (z1 , ¯z1 )VkL kR (z2 , ¯z2 ) ∼ z12 L

. (8.2.19) One encounters branch cuts in various expressions, which is not surprising because the ﬁeld X is no longer single-valued. It is important, however, that the full vertex operator OPE is single-valued: the net phase when z1 circles z2 is exp[πiα (kL kL − kR kR )] = exp[2πi(nw + wn )] = 1 .

(8.2.20)

This is as it must be in order for the string amplitudes to be well deﬁned.

239

8.2 Toroidal compactiﬁcation in CFT A technicality

The above is essentially correct, but we have been cavalier with the location of the branch cut in the logarithm. One can see a problem in the OPE (8.2.19) if one interchanges the points z1 and z2 and the momenta k and k . The left-hand side is symmetric, but the right-hand side changes by exp[πi(nw + wn )] and so changes sign if nw + wn is odd. One can also see the problem as follows. From the mode expansion one may derive the equal time (|z1 | = |z2 |) commutator [ XL (z1 ), XL (z2 ) ] =

πiα sign(σ11 − σ21 ) . 2

(8.2.21)

The CBH formula then shows that if we deﬁne the operators simply as creation–annihilation ordered, then for nw + wn odd the operators VkL kR and VkL kR anticommute rather than commute; the invisible branch cut (8.2.20) is separated into two visible ones. A correct oscillator expression for the vertex operator (there is some arbitrariness in the phase) is VkL kR (z, ¯z ) = exp[πi(kL − kR )(pL + pR )α /4] ◦◦ eikL XL (z)+ikR XR (¯z ) ◦◦ ,

(8.2.22)

where as usual the ps are momentum operators and the ks are numbers (the momenta carried by the given vertex operator). When VkL kR and VkL kR are commuted past each other, the additional factors, known as cocycles, give an additional phase #

exp πi (kL − kR )(kL + kR ) − (kL − kR )(kL + kR ) α /4

$

= exp[πi(nw − wn )] ,

(8.2.23)

removing the branch cuts. For most purposes one can ignore this complication and work with the simpler expressions of the previous paragraph; the cocycle aﬀects only the relative signs of certain amplitudes. The general X µ path integral (6.2.18) factorizes in an obvious way into holomorphic times antiholomorphic, so that one can simply replace n

|zij |α ki kj →

i,j=1 i<j

and also replace the 2πδ(

n α kLi kLj /2 α kRi kRj /2

zij

¯zij

,

(8.2.24)

i,j=1 i<j

k) from the noncompact integral over x0 by 2πRδΣi ni ,0 δΣi wi ,0 .

(8.2.25)

Again this is naive; the precise expression (8.2.22) gives some additional signs. The reader can readily calculate an amplitude with four vertex operators, which is similar to the Virasoro–Shapiro amplitude.

240

8 Toroidal compactiﬁcation and T -duality DDF operators

We have just seen that the exponential operator can be separated into holomorphic and antiholomorphic parts. This has many applications. We now use it to tie up a loose end from the end of chapter 4, the DDF operators. We will work with light-cone coordinates X ± = 2−1/2 (X 0 ± X 1 ), whose holomorphic parts have the OPE (we drop the subscript L) X + (z)X − (0) ∼

α ln z , 2

X + (z)X + (0) ∼ X − (z)X − (0) ∼ 0 .

(8.2.26)

It follows that the operators V i (nk0 , z) = ∂X i (z)eink0 X

+ (z)

(2/α )1/2

(8.2.27)

with i transverse are (1, 0) primary ﬁelds with OPE V i (nk0 , z) V j (mk0 , 0) ∼ − −

δ ij i(n+m)k0 X + (0) e z2

+ ink0 δ ij ∂X + (0)ei(n+m)k0 X (0) . z

(8.2.28)

Deﬁne the DDF operators by

Ain

=

dz i V (nk0 , z) . 2π

(8.2.29)

The residue of the z −1 term in the OPE (8.2.28) is a total derivative except when n + m = 0, and so the DDF operators satisfy an oscillator algebra [Aim , Ajn ] = mδ ij δm,−n

α k0 p+ , 2

(8.2.30)

which is an oscillator algebra, up to normalization. Consider the action of the DDF operators on a state of given momentum q. The OPE of V i (nk0 , z) with the vertex operator for this state − involves z −α nk0 q /2 and so is single-valued if we ﬁx k0 = 2/α q − . The contour integral (8.2.29) then deﬁnes a sensible operator in this sector. The DDF operators, being integrals of (1, 0) tensors, commute with the Virasoro generators and so take physical states into physical states. It follows that we can build physical states by acting with the DDF operators on the oscillator ground state. The states obtained in this way are in one-to-one correspondence with the light-cone states, and so are a complete set of physical states. The DDF operators do not involve X − and so these states have no α− −m excitations. They are therefore the same states given by the more general but less explicit construction (4.4.19).

8.3 Closed strings and T -duality 8.3

241

Closed strings and T -duality

Now that we are looking at string theory we take D = 26, with only X 25 ˜ 0 ) conditions are periodic at ﬁrst. The mass-shell (L0 and L 4 (N − 1) α 4 ˜ = (kR25 )2 + (N − 1) , α

m2 = −k µ kµ = (kL25 )2 +

(8.3.1)

or n2 w2R2 2 ˜ − 2) , + + (N + N 2 2 R α α ˜ . 0 = nw + N − N

m2 =

(8.3.2a) (8.3.2b)

One recognizes four contributions to the mass-squared: those from the compact momentum, the potential energy of the winding string, the oscillators, and the zero-point energy. The discussion of the physical spectrum in chapter 4 was valid for an arbitrary compactiﬁcation, so we get the correct counting of states by considering only the transverse oscillators M = 2, . . . , 25. Let us ﬁrst recover the ﬁeld theory results for the massless spectrum. At a generic value of R, the only way a state can be massless is if n = w = 0 ˜ = 1. These are the same 242 states as in the noncompact and N = N theory, but it is useful now to separate them according to whether the oscillations are in the spacetime directions µ or the internal direction 25: αν−1 |0; k , αµ−1 ˜

25 µ (αµ−1 ˜α25 α−1 )|0; k , −1 + α−1 ˜

25 µ α25 α−1 )|0; k , (αµ−1 ˜ −1 − α−1 ˜

α25 α25 −1 ˜ −1 |0; k .

(8.3.3)

The ﬁrst of these breaks up further into a 25-dimensional graviton plus dilaton plus antisymmetric tensor. The second, a graviton with one spacetime and one internal index, is the Kaluza–Klein vector. The third state is the vector from the antisymmetric tensor. The ﬁnal state is a scalar, the modulus for the radius of the compact direction. Its vertex operator, ¯ 25 eik·X :, is a perturbation of the metric G25,25 . This is the same : ∂X 25 ∂X massless spectrum as found by considering the low energy ﬁeld theory. It is interesting to consider the charges of massive states under the U(1) × U(1) symmetry. Again the Kaluza–Klein gauge symmetry is from X 25 -translations, so the corresponding charge is the compact momentum p25 . For the antisymmetric tensor gauge symmetry, consider the zeromomentum vertex operator, which measures the charge of any state to which it couples. It is proportional to ¯ 25 − ∂X 25 ∂X ¯ µ = ∂(X ¯ µ) . ¯ 25 ∂X µ ) − ∂(X 25 ∂X ∂X µ ∂X

(8.3.4)

242

8 Toroidal compactiﬁcation and T -duality

This is a total derivative, but it does not integrate to zero in a winding state because X 25 is not single-valued. So the Bµ,25 charge is the winding number. This is our ﬁrst of several examples of ‘stringy’ physics in this compactiﬁcation: in ﬁeld theory there were no states carrying this charge. Let us check the gauge couplings explicitly from the string three-point coupling. The gauge boson vertex operators are 21/2 gc,25 ¯ 25 ± ∂X 25 ∂X ¯ µ )eik·X : . : (∂X µ ∂X (8.3.5) α We have deﬁned the 25-dimensional string coupling gc,25 = gc (2πR)−1/2 . The factor (2πR)−1/2 is from normalizing the zero-mode wavefunction. The vertex operator for a tachyon with general compact momentum and winding is gc,25 : eikL ·XL (z)+ikR ·XR (¯z ) : .

(8.3.6)

The three-point amplitude for one gauge boson and two tachyons, with the tachyons having nonzero compact momenta and winding, is similar to eq. (6.6.14), −2−1/2 πigc,25 (2π)25 δ 25 ( → −23/2 πigc,25 (2π)25 δ 25 (

µ 25 25 i ki )k23 (kL23 ± kR23 ) µ 25 25 i ki )k2 (kL2 ± kR2 ) .

(8.3.7)

In the second line we have taken the gauge boson momentum k1 → 0, which deﬁnes the gauge coupling. The two gauge bosons thus couple 25 ±k 25 , the compact momentum and winding, as expected. respectively to kL2 R2 By considering also the amplitude with a graviton replacing the gauge boson, one can recover the same relations (8.1.11), (8.1.12) between the various couplings as found from the eﬀective action. Enhanced gauge symmetries Everything so far is almost exactly as it would be in ﬁeld theory with one compact dimension. The 26-dimensional graviton gives rise to a 25dimensional graviton plus vector plus modulus, and so on. The one stringy eﬀect is that winding states carry the Bµ,25 charge. Further and more striking eﬀects arise at special compactiﬁcation radii. Our discussion of the massless spectrum omitted states that are massless only for special values of the radius R. The richest case is R = α 1/2 , where 25 = (n ± w)α−1/2 and the condition for a massless state is kL,R ˜ = 4. (n + w)2 + 4N = (n − w)2 + 4N (8.3.8) ˜ = 1, there are now In addition to the generic solution n = w = 0, N = N also ˜ = 1 , n = −w = ±1, N = 1, N ˜ = 0 , (8.3.9) n = w = ±1, N = 0, N

8.3 Closed strings and T -duality

243

and ˜ =0, n = ±2, w = N = N

˜ =0. w = ±2, n = N = N

(8.3.10)

The states (8.3.9) include four new gauge bosons, with vertex operators ¯ µ eik·X exp[±2iα−1/2 X 25 ] : , : ∂X µ eik·X exp[±2iα−1/2 X 25 ] : . (8.3.11) : ∂X L R The precise deﬁnition of the exponential operators is as discussed in the previous section. These states have internal momentum and winding number, so they carry the Kaluza–Klein and antisymmetric tensor gauge charges. The only consistent theory of charged massless vectors is nonAbelian gauge theory, so the new gauge bosons must combine with the old into a non-Abelian theory. It is now useful to work with the basis ¯ µ and ∂X µ ∂X ¯ 25 for the earlier vectors. The ﬁrst of these couples to ∂X 25 ∂X 25 kL , under which the ﬁrst pair of states in (8.3.11) carries charges ±1 and the second pair is neutral. The second similarly couples to kR25 and so to the other pair. This identiﬁes the gauge group as S U(2) × S U(2), with the ¯ µ forming one S U(2) and the three involving three vectors involving ∂X µ ∂X forming the other. To exhibit the S U(2) × S U(2) deﬁne the three (1, 0) currents j 1 (z) = : cos[2α−1/2 XL25 (z)] : , 2

j (z) = : sin[2α 3

j (z) =

−1/2

i∂XL25 (z)/α

XL25 (z)] : 1/2 .

,

(8.3.12a) (8.3.12b) (8.3.12c)

These have been normalized to have the OPE δ ij ijk k +i (8.3.13) j (0) . 2 2z z There is of course also a similar OPE of (0, 1) currents ˜ i . The single pole term implies that the corresponding charges form an S U(2) algebra. In fact, since the currents are holomorphic there is an inﬁnite-dimensional algebra formed by the Laurent coeﬃcients j i (z)j j (0) ∼

i

j (z) =

∞

jmi

m=−∞

z m+1

,

(8.3.14a)

m k , (8.3.14b) δm,−n δ ij + iijk jm+n 2 known variously as a current algebra, aﬃne Lie algebra, or Kac–Moody algebra. We will encounter such algebras extensively beginning in chapter 11. It will be shown that the coeﬃcient of the z −2 term is quantized, with the value in (8.3.13) being the minimum, so this is known as a level one S U(2) current algebra. The existence of this S U(2) × S U(2) symmetry is the ﬁrst indication that string theory sees spacetime geometry diﬀerently from what we are used [ jmi , jnj ] =

244

8 Toroidal compactiﬁcation and T -duality

to in ﬁeld theory, where only a U(1) × U(1) symmetry is evident at any radius. The j 3 and j 1,2 currents appear to have very diﬀerent origins, as oscillator excitations and as winding–momentum excitations respectively, and their expressions in terms of X 25 are very diﬀerent, but in their action on the string spectrum they are related to one another by symmetry. The extra vectors are massless because the energies of internal momentum and winding are oﬀset by the negative zero-point energy. The reader might therefore suspect that these could arise only in theories with tachyons, but we will see otherwise in volume two. Scales and couplings At the S U(2) × S U(2) radius, the relation (8.1.11) becomes2 2 g25 = 2κ225 /α .

(8.3.15)

This relation continues to hold if we compactify more dimensions, each side scaling the same way, so in particular in four dimensions g42 = 2κ24 /α .

(8.3.16)

In nature, the non-Abelian gauge couplings g42 are within an order of magnitude of unity, so this implies that the string length α1/2 is not too far from the gravitational length κ4 . We can also think about this as follows. The four-dimensional gauge coupling is dimensionless, but the four-dimensional gravitational coupling is dimensional. Deﬁne an eﬀective dimensionless gravitational coupling depending on the energy scale E, 2 (E) = κ24 E 2 , gG,4

(8.3.17)

which increases quadratically with energy. Of course, the ‘dimensionless’ gauge couplings run with energy, but this is slow compared to the dimensional scaling of the gravitational coupling. Then by eq. (8.3.16), the string mass scale is where the gauge and gravitational couplings are roughly 2 (E) ≈ g 2 . equal, gG,4 4 It is also interesting to consider the possibility that in compactifying down to four dimensions, some of the compactiﬁed dimensions are larger than others. This can best be analyzed by starting at low energy and working upwards. At low energy there is a dimensionless gauge coupling g42 . At an energy ρ−1 5 , one compact dimension becomes visible and the physics 2

To be precise, this was the coupling of an (n, w) = (1, 0) state. The non-Abelian gauge bosons have (|n|, |w|) = (1, 1) and so their coupling, which is the conventional particle physics deﬁnition 2 = 4κ225 /α . We will see in chapter 18 that this holds for all level of the SU(2) coupling, is gSU(2),25 one current algebras.

8.3 Closed strings and T -duality

245

becomes ﬁve-dimensional (for simplicity we consider only one dimension at this scale; with more dimensions the conclusion is strengthened). The eﬀective ﬁve-dimensional coupling is g52 = 2πρ5 g42 .

(8.3.18)

This follows from the eﬀective action, as in eq. (8.1.12). The coupling g52 has units of length. That is, ﬁve-dimensional Yang–Mills theory, like four-dimensional gravity, is a nonrenormalizable theory. Correspondingly, the eﬀective dimensionless coupling gˆ 52 = g52 E = 2πρ5 Eg42

(8.3.19)

grows linearly with energy. However, g42 is not much less than one in nature, so the coupling quickly becomes strong at high energy; presumably string theory is then needed to make the short-distance behavior sensible. So the compactiﬁcation scale must be close to the string scale, which as we have already learned is close to the gravitational scale. This analysis is modiﬁed somewhat in open string theory, and when the recent understanding of strongly coupled string theory is taken into account; it is possible, but unlikely, that string or Kaluza–Klein excitations lie far below the Planck scale. Higgs mechanism It is instructive to consider what happens as we move R away from the S U(2) × S U(2) radius. The extra gauge bosons now acquire masses m=

2 |R 2 − α | 1/2 ≈ R − α , Rα α

(8.3.20)

the approximation being for radii close to the S U(2) × S U(2) radius α1/2 . Close to this radius the mass is much less than the string scale and we should be able to understand the physics in terms of low energy ﬁeld theory. There is only one way to give such a mass to a gauge boson, spontaneous symmetry breaking, and that is just what is happening here. At R = α 1/2 there are ten massless scalars: the dilaton, the modulus G25,25 , the four states in (8.3.9) where a 25-oscillator is excited, and the four states (8.3.10). The last nine are created by the vertex operators : j i (z)˜ j (¯z )eik·X(z,¯z ) : .

(8.3.21)

The index i is a vector under the left-moving S U(2) and the index j is a vector under the right-moving S U(2). That is, these transform as (3,3) under S U(2) × S U(2). In particular, the modulus for the radius is j 3˜ 3 . Moving away from the S U(2) × S U(2) radius gives an expectation value

246

8 Toroidal compactiﬁcation and T -duality

to this ﬁeld and so breaks the gauge symmetry down to the U(1) × U(1) that leaves the 3-axes invariant. Indeed, this is the gauge group at generic radii. Near the S U(2) radius, the mass (8.3.20) is linear in the magnitude |R − α1/2 | of the modulus, as usual in spontaneous breaking. Let us label the spacetime ﬁelds corresponding to the nine massless scalars as Mij . A change in the modulus is M33 = 0, but it is instructive to look for more general backgrounds near the S U(2) × S U(2) point. The interactions of these massless ﬁelds will be described by some spacetime action, which will include a potential U(M). These ﬁelds are massless at the symmetric point so there is no term of order M 2 in U, but an (S U(2) × S U(2))-invariant cubic term is possible,

U(M) ∝ ijk i j k Mii Mjj Mkk = det M .

(8.3.22)

It is not hard to verify from the string three-point amplitude that this term does appear. Being cubic in bosons it is unbounded below, but like the tachyon this is an irrelevant artifact of bosonic string theory. It is interesting to look for static classical solutions, ignoring the fact that they are unstable to small variations. In order to have a static background solution, the graviton and dilaton ﬁeld equations require the potential to vanish and the Mij ﬁeld equations require the potential to be stationary, U(M) =

∂U(M) =0. ∂Mij

(8.3.23)

Using an S U(2) × S U(2) rotation to diagonalize M, the conditions (8.3.23) become M11 M22 M33 = M11 M22 = M11 M33 = M22 M33 = 0 .

(8.3.24)

Thus, only one of the diagonal components can be nonvanishing. By an S U(2)×S U(2) rotation we can take this to be M33 . So we do not construct any physically new string backgrounds in this way. A continuous family of static background solutions is known as a ﬂat direction. Here there are nine massless scalars, which by an S U(2) × S U(2) rotation are reduced to the three diagonal ﬁelds, but there is only one ﬂat direction (gauge-equivalent directions not being counted). Of course, we have only computed the cubic term in the potential, and in a general situation this could vanish and higher powers still be nonzero. To establish an exactly ﬂat direction one needs some special argument. Here the special argument is that we can explicitly construct the free CFT for any value of R. Notice that if we treat j 1˜ 1 or j 2˜ 2 as the modulus the world-sheet action looks very complicated, with cosines and sines of X 25 , but the world-sheet theory is solvable because it is equivalent to the free theory obtained by varying R.

8.3 Closed strings and T -duality

247

T -duality From the mass formula n2 w2R2 2 ˜ − 2) , + + (N + N (8.3.25) R2 α2 α we see that as R → ∞ winding states become inﬁnitely massive, while the compact momenta go over to a continuous spectrum. This is as would be expected with a noncompact dimension. Looking at the opposite limit R → 0, however, one gets a surprise. The states with compact momentum become inﬁnitely massive, but the spectrum of winding states now approaches a continuum — it does not cost much energy to wrap a string around a small circle. Thus as the radius goes to zero the spectrum again seems to approach that of a noncompact dimension. This is quite diﬀerent from the behavior in ﬁeld theory, where there is compact momentum n but no winding number w, and no states become light as R → 0. In fact, the R → 0 and R → ∞ limits are physically identical. The spectrum (8.3.25) is invariant under m2 =

α , n↔w . (8.3.26) R This equivalence extends to the interactions as well. Notice that reversing n and w is the same as R → R =

25 p25 L → pL ,

25 p25 R → −pR .

(8.3.27)

Consider the theory at radius R. Recalling the separation X 25 (z, ¯z ) = XL25 (z) + XR25 (¯z ), deﬁne X 25 (z, ¯z ) = XL25 (z) − XR25 (¯z ) .

(8.3.28)

The ﬁeld X 25 has the same OPEs and energy momentum tensor as X 25 , the minus signs always entering in pairs. The only change in the CFT from using X 25 in place of X 25 is that it includes the sign change (8.3.27), which changes the spectrum from that of the theory with radius R to that of the theory with radius R . That is, these are the same theory, one written in terms of X 25 and one in terms of X 25 . It should be noted that the relation between X 25 (z, ¯z ) and X 25 (z, ¯z ) is nonlocal on the world-sheet, because the separate left- and right-moving ﬁelds are nonlocal operators. This equivalence is known as T -duality. That the R → 0 limit is physically the same as the R → ∞ limit is completely unlike the behavior of point particles, and is one more sign that strings see geometry very diﬀerently at short distances. The space of inequivalent theories is the half-line R ≥ α1/2 . We could take instead the range 0 ≤ R ≤ α1/2 but it is more natural to think in terms of the larger of the two equivalent radii:

248

8 Toroidal compactiﬁcation and T -duality

momentum continua are more familiar than winding number continua, and in particular questions of locality are clearer in the larger-R picture. Thus, there is no radius smaller than the self-dual radius R self-dual = RSU(2)×SU(2) = α1/2 .

(8.3.29)

This indication of a minimum distance scale of the order of the string length will recur elsewhere in string perturbation theory, but we will see that nonperturbatively there is structure at shorter distance. It is important in many applications that T -duality acts nontrivially on the string dilaton Φ. Consider the scattering amplitude for gravitons without winding or compact momentum. These states are invariant under T -duality, and so must be the amplitude as well. The latter can be read from the low energy action (8.1.9), and so the 25-dimensional coupling κ25 must be invariant under duality. Since the 26-dimensional coupling is related to this by κ = (2πρ)1/2 κ25 , duality must act as ρ =

α , ρ

κ =

α1/2 κ. ρ

(8.3.30)

Since κ ∝ eΦ , this implies

eΦ =

α1/2 Φ e . ρ

(8.3.31)

T -duality is a symmetry, relating diﬀerent states (backgrounds) in a single theory, and in fact it is a gauge symmetry. We saw that the modulus δR is the 33-component of a (3,3) ﬁeld. A rotation by π around the 1-axis in one of the S U(2)s reverses the sign of this modulus, so decreasing R from the S U(2) × S U(2) radius is gauge-equivalent to increasing it. Thus the Z2 symmetry of duality is a small part of the S U(2) × S U(2) gauge symmetry. This implies further that duality is a symmetry not only of string perturbation theory but of the exact theory. If we have a massless gauge boson to leading approximation, even small explicit violations of the symmetry lead to inconsistency (spontaneous violation is not an issue, as T -duality is already spontaneously broken away from the self-dual radius). This last argument illustrates an important idea, the complementary use of stringy and spacetime arguments. The existence of the enhanced gauge symmetry is a purely stringy phenomenon. We do not have a nonperturbative understanding of string theory, but we do know a great deal about low energy ﬁeld theory, even nonperturbatively, and we can use this.

8.4 Compactiﬁcation of several dimensions 8.4

249

Compactiﬁcation of several dimensions

It is useful to generalize the analysis to k periodic dimensions, Xm ∼ = X m + 2πR , 26 − k ≤ m ≤ 25 .

(8.4.1)

Let d = 26 − k be the number of noncompact dimensions. Spacetime is now M d ×T k . The coordinate periodicity (8.4.1) is being held ﬁxed, but the actual geometry of the k-torus depends on internal metric Gmn . With more than one compact dimension, the antisymmetric tensor also has scalar components Bmn . The total number of scalars from both sources is k 2 . There are also Kaluza–Klein gauge bosons Am µ and antisymmetric tensor gauge bosons Bmµ . The low energy action, obtained as in section 8.1, is

S =

(2πR)k dd x (−Gd )1/2 e−2Φd Rd + 4∂µ Φd ∂µ Φd 2κ20 1 − Gmn Gpq (∂µ Gmp ∂µ Gnq + ∂µ Bmp ∂µ Bnq ) 4 1 1 1 m nµν − Gmn Fµν F − Gmn Hmµν Hnµν − Hµνλ H µνλ , 4 4 12

(8.4.2)

where Φd = Φ − 14 ln det Gmn . The string spectrum The main new issue is the antisymmetric tensor background Bmn . The contribution of this term to the world-sheet Lagrangian density is proportional to Bmn ∂a (g 1/2 ab X m ∂b X n ) ,

(8.4.3)

which is a total derivative for constant Bmn . It thus has no eﬀect locally and the world-sheet theory is still a CFT, a good string background. It does, however, change the spectrum. We will see this in two ways, canonically and via the path integral. In the canonical approach, focus on the zero-mode contribution to the world-sheet action. Inserting X m (σ 1 , σ 2 ) = xm (σ 2 ) + w m Rσ 1

(8.4.4)

into the world-sheet action gives i 1 ˙n + w m w n R 2 ) − Bmn x ˙m w n R . xm x (8.4.5) L = Gmn (˙ 2α α A dot indicates a derivative with respect to world-sheet time σ 2 . The canonical momenta are ∂L 1 pm = − m = (Gmn v n + Bmn w n R) , (8.4.6) ∂v α

250

8 Toroidal compactiﬁcation and T -duality

where v m = i˙ xm . Some unfamiliar signs and is appear because we are using Euclidean world-sheet time. Continuing to Minkowski time, v m becomes the velocity ∂0 xm . The periodicity of the wavefunction implies the quantization of the canonical momenta, pm = nm /R, and so nm vm = α (8.4.7) − Bmn w n R . R The zero-mode contribution to the world-sheet Hamiltonian is 1 Gmn (v m v n + w m w n R 2 ) , (8.4.8) 2α and the closed string mass is 1 2 ˜ − 2) , Gmn (vLm vLn + vRm vRn ) + (N + N 2α2 α = vm ± wmR .

m2 = m vL,R

(8.4.9a) (8.4.9b)

The Bmn background thus shifts the masses of the winding states through ˜ 0 constraint is the dependence of v m on Bmn . The L0 − L ˜ 0 = Gmn (vLm vLn − vRm vRn ) + 4α (N − N) ˜ . = 4α (nm w m + N − N)

(8.4.10)

Alternatively, consider the torus path integral for the partition function. The antisymmetric tensor term is locally a total derivative and so depends only on the topology of the conﬁguration. Consider the conﬁguration X m = (w1m σ 1 + w2m σ 2 )R .

(8.4.11)

The spatial direction on the torus winds w1m times around each periodic spacetime dimension xm , and the timelike dimension w2m times. The Bmn world-sheet action is 2πibmn w1m w2n ,

(8.4.12)

where bmn = Bmn R 2 /α . One can check that the sum over path integral sectors with phase (8.4.12) is related by Poisson resummation to the partition function for the shifted spectrum (8.4.9). In the description above the metric Gmn appears in the action for the world-sheet ﬁelds. For vertex operator calculations it is more convenient to keep the usual action. Write the metric in terms of a tetrad emr , Gmn = emr enr ,

(8.4.13)

with tangent space indices r, s, . . . . The coordinates X r = emr X m then have the standard OPEs. The vertex operator momenta are vmL vmR (8.4.14) krL = erm , krR = erm , α α

8.4 Compactiﬁcation of several dimensions with the inverse tetrad erm . The mass-shell conditions become 1 2 ˜ − 2) , m2 = (krL krL + krR krR ) + (N + N 2 α ˜ . 0 = α (krL krL − krR krR ) + 4(N − N)

251

(8.4.15a) (8.4.15b)

We use the coordinates X r implicitly in any discussion of vertex operators, such as the one that now follows. Narain compactiﬁcation There is an elegant description of the general toroidal compactiﬁcation. Consider the winding state vertex operators eikL ·XL +ikR ·XR . For any given compactiﬁcation, the spectrum of momenta (krL , krR ) forms a lattice in a 2k-dimensional momentum space R2k . That is, the momentum spectrum consists of all integer linear combinations of 2k linearly independent basis vectors. Let us work with the dimensionless momenta lL,R = kL,R (α /2)1/2 and call the corresponding lattice Γ. The OPE of two vertex operators is

: eikL ·XL (z)+ikR ·XR (¯z ) : : eikL ·XL (0)+ikR ·XR (0) : ∼ z lL ·lL ¯z lR ·lR : ei(kL +kL )·XL (0)+i(kR +kR )·XR (0) : .

(8.4.16)

As one vertex operator circles the other, the product picks up the phase exp[2πi(lL · lL − lR · lR )]. Single-valuedness of the operator product thus requires lL · lL − lR · lR ≡ l ◦ l ∈ Z

(8.4.17)

l

for all l and in Γ. We have deﬁned the product ◦, which has signature (k, k) in R2k . The dual lattice Γ∗ is deﬁned to be the set of points in R2k that have integer ◦ product with all points in Γ. The single-valuedness condition (8.4.17) is then the statement that Γ ⊂ Γ∗ .

(8.4.18)

Modular invariance also constrains Γ. In section 7.2 it was shown that ˜ 0 be an integer for all invariance under τ → τ + 1 requires that L0 − L states in the string CFT. Then l ◦ l ∈ 2Z for all l ∈ Γ .

(8.4.19)

The condition (8.4.19) actually implies condition (8.4.17), given closure of the OPE 2l ◦ l = (l + l ) ◦ (l + l ) − l ◦ l − l ◦ l ∈ 2Z .

(8.4.20)

Modular invariance under τ → −1/τ requires a more detailed calculation. The partition function for the compact dimensions is ZΓ (τ) = |η(τ)|−2k

l∈Γ

exp(πiτlL2 − πi¯τlR2 ) .

(8.4.21)

252

8 Toroidal compactiﬁcation and T -duality

To determine the transformation property of ZΓ use again Poisson resummation. Write

δ(l − l ) = VΓ−1

l ∈Γ

exp(2πil ◦ l) .

(8.4.22)

l ∈Γ∗

Here, VΓ is the volume of a unit cell of the lattice Γ. Eq. (8.4.22) is familiar from Fourier series: the phase averages to zero except when l ∈ Γ, and the normalization is ﬁxed by integrating over a unit cell. Using the representation (8.4.22), the sum (8.4.21) can be written ZΓ (τ) = VΓ−1 |η(τ)|−2k =

l ∈Γ∗ −1 −k/2 VΓ (τ¯τ) |η(τ)|−2k

d2k l exp(2πil ◦ l + πiτlL2 − πi¯τlR2 )

exp(−πilL2 /τ + πilR2 /¯τ)

l ∈Γ∗

= VΓ−1 ZΓ∗ (−1/τ) ,

(8.4.23)

where in the last line we have used the modular transformation (7.2.44) of the eta function. Then the condition Γ = Γ∗

(8.4.24)

is suﬃcient for modular invariance, because VΓ = VΓ−1 ∗ and so is equal to unity if (8.4.24) holds. A little thought shows that this is also necessary if modular invariance is to hold for all τ. The consistency conditions can be summarized as the requirement that Γ be an even self-dual lattice of signature (k, k), even referring to the property (8.4.19) and self-dual to the property (8.4.24). All such lattices have been classiﬁed. Note that the consistency conditions (8.4.19) and (8.4.24) depend on the momenta l only through the ◦ product, which is invariant under Lorentz boosts of the 2k-dimensional space, O(k, k, R) transformations. If Γ is an even self-dual lattice then so is Γ = ΛΓ ,

(8.4.25)

which is obtained by rotating all the points of the lattice by a common O(k, k, R) transformation Λ. Now, it is important that O(k, k, R) is not a symmetry of the theory. The mass-shell condition and operator products involve the separate dot products lL · lL and lR · lR and so are invariant only under the O(k, R) × O(k, R) that acts on lL and lR separately. Thus, most O(k, k, R) transformations produce inequivalent theories. Consider for example k = 1, where n mr (8.4.26) lL,R = ± r 2

8.4 Compactiﬁcation of several dimensions

253

in terms of the dimensionless radius r = R(2/α )1/2 . This is indeed an even self-dual lattice. The boost lL = lL cosh λ + lR sinh λ ,

lR = lL sinh λ + lR cosh λ

(8.4.27)

changes this to the lattice for r = re−λ . All even self-dual lattices of given Lorentzian signature can be obtained from any single lattice by O(k, k, R) transformations. Start with a given solution, for example all compact dimensions orthogonal and at the S U(2) radius with Bmn = 0. The corresponding momentum lattice Γ0 is known as kP2 and gives the gauge group S U(2)2k . As discussed above, two O(k, k, R) transformations Λ and Λ Λ give equivalent string theories if Λ ∈ O(k, R) × O(k, R), so the space of inequivalent theories obtained in this way is O(k, k, R) , O(k, R) × O(k, R)

(8.4.28)

up to some discrete identiﬁcations to be discussed shortly. This is equivalent to the earlier description in terms of the backgrounds of Gmn and Bmn . In particular, the number of parameters in the coset (8.4.28) is 2k(2k − 1) (8.4.29) − k(k − 1) = k 2 , 2 the same as the number of components in Gmn and Bmn . The T -duality symmetry is much enlarged from the case of one compactiﬁed dimension. In the abstract description, there is some discrete subgroup of O(k, k, R) that takes the initial lattice Γ0 into itself (though it permutes the individual points of the lattice). This subgroup is conventionally but somewhat inaccurately denoted O(k, k, Z). If Λ is in this group, then clearly ΛΓ0 and ΛΛ Γ0 are the same lattice. In all we have the equivalence ΛΓ0 ∼ = Λ ΛΛ Γ0 , Λ ∈ O(k, R) × O(k, R) ,

Λ ∈ O(k, k, Z) .

(8.4.30a) (8.4.30b)

The space of inequivalent lattices, and inequivalent backgrounds, is then O(k, k, R) . O(k, R) × O(k, R) × O(k, k, Z)

(8.4.31)

One should remember that the continuous denominator group acts by left-multiplication and the discrete group by right-multiplication, as in eq. (8.4.30a). In terms of the backgrounds, the T -duality group O(k, k, Z) contains transformations of several types. One is R → α /R dualities on the individual axes. Another is large spacetime coordinate transformations respecting

254

8 Toroidal compactiﬁcation and T -duality

the periodicity, xm = Lmn xn

(8.4.32)

with Lmn integers and det L = 1 for invertibility. This is the group S L(k, Z). Finally there are shifts of the antisymmetric tensor background, bmn → bmn + Nmn

(8.4.33)

with Nmn integers. From either the canonical result (8.4.7) or the path integral phase (8.4.12) it follows that this leaves the spectrum invariant. Together these generate the whole T -duality group. Identifying ﬁrst by the left-multiplication produces the space (8.4.28), and one can think of the discrete right-multiplication as acting on this covering space. For generic Λ, there are no Λ1,2 in the respective denominator groups such that Λ1 ΛΛ2 = Λ, so that there are no T -dualities that leave this point on the covering space ﬁxed. At special Λ there are solutions to Λ1 ΛΛ2 = Λ, and these will be ﬁxed points of the T -duality element Λ2 . In the one-dimensional case we had such a ﬁxed point, which was also a point of enhanced gauge symmetry. In the higher-dimensional case, besides products of S U(2)s, larger gauge groups arise at various points. We will defer further discussion of this to chapter 11. When there is a moduli space parameterized by scalar ﬁelds φi , the kinetic term 1 − gij (φ)∂µ φi ∂µ φj 2

(8.4.34)

deﬁnes a natural metric on moduli space. To be precise, an unambiguous deﬁnition requires that one ﬁrst perform a spacetime Weyl transformation as in eq. (3.7.25) so as to make the coeﬃcient of the gravitational action moduli-independent and eliminate mixing between the moduli and the spacetime metric. In the low energy action (8.4.2), the resulting scalar kinetic terms are proportional to 16 ∂µ Φ∂µ Φ + Gmn Gpq (∂µ Gmp ∂µ Gnq + ∂µ Bmp ∂µ Bnq ) . d−2

(8.4.35)

The coset space (8.4.31) has a unique O(k, k, R)-invariant metric, which is just the second term in eq. (8.4.35). This O(k, k, R) is not a symmetry of the full theory; only its discrete T -duality subgroup O(k, k, Z) is. The diﬀerence between the two is not visible at low energy — it comes from the quantization of the string zero modes and so aﬀects the massive spectrum. Thus O(k, k, R) is an accidental symmetry of the low energy theory. In this example the moduli space is a product of the dilaton moduli space and the moduli space of compactiﬁcations.

8.4 Compactiﬁcation of several dimensions

255

An example The case of two compact dimensions is a good example. The four moduli G24,24 , G24,25 , G25,25 , and B24,25 are usefully joined into two complex ﬁelds τ = τ1 + iτ2 and ρ = ρ1 + iρ2 . Deﬁne R2 B24,25 + i det 1/2 Gmn α V = b24,25 + i 2 , (8.4.36) 4π α where V is the volume of the compact two-torus. Parameterize the compact metric as 2 α ρ2 ds2 = 2 dX 24 + τdX 25 . (8.4.37) R τ2

ρ=

The coordinate transformations (8.4.32) act on the spacetime two-torus just as the modular group acts on the world-sheet two-torus, generating a P S L(2, Z) on τ and leaving ρ invariant. The antisymmetric tensor shift (8.4.33) is ρ → ρ + N24,25 with τ invariant. A simultaneous T -duality on X 24,25 acts as ρ → −1/ρ with τ invariant. Together the last two transformations generate a second P S L(2, Z), acting on ρ. In addition, duality on X 24 alone takes (τ, ρ) → (ρ, τ) and spacetime parity X 24 → −X 24 takes (τ, ρ) → (−¯τ, −¯ ρ). The full T -duality group is then P S L(2, Z) × P S L(2, Z)× Z22 .

(8.4.38)

A further equivalence comes from world-sheet parity, which takes (τ, ρ) → (τ, −¯ ρ). It reverses the sign of the ◦ product. The moduli space is therefore two copies of the moduli space of the torus, together with some discrete identiﬁcations. The kinetic terms are proportional to ¯ ∂µ τ∂µ¯τ ∂µ ρ∂µ ρ + . 2 2 τ2 ρ2

(8.4.39)

The modular-invariant combination d2 τ/τ22 was encountered in the discussion of the genus-zero amplitude. The continuous group P S L(2, R) carries any τ in the upper half-plane into any other, and a U(1) subgroup leaves each τ invariant. Thus the upper half-plane can be regarded as P S L(2, R)/U(1), and the full moduli space is P S L(2, R) × P S L(2, R) . U(1) × U(1) × P S L(2, Z) × P S L(2, Z) × Z22

(8.4.40)

This is the same as the earlier (8.4.31). In particular, P S L(2, R)×P S L(2, R) is locally the same as O(2, 2, R), in parallel to the fact that S U(2) × S U(2) is locally the same as O(4, R).

256

8 Toroidal compactiﬁcation and T -duality 8.5

Orbifolds

Rather than the periodic identiﬁcation of X 25 , consider instead identifying points under the reﬂection X 25 ∼ = −X 25 .

(8.5.1)

Now a fundamental region is the half-line X 25 ≥ 0, and the hyperplane X 25 = 0 becomes a boundary. This hyperplane is distinguished by the fact that its points are ﬁxed under the reﬂection (8.5.1). More generally we could identify spacetime under the simultaneous reﬂection of k coordinates X m → −X m ,

26 − k ≤ m ≤ 25 .

(8.5.2)

Again there is a space of ﬁxed points, at X 26−k = . . . = X 25 = 0. For k ≥ 2 this space is a singularity. In particular, for k = 2 it is a conic singularity with deﬁcit angle π. One can form a compact space by combining the reﬂection (8.5.1) with the periodic identiﬁcation X 25 ∼ = X 25 + 2πR; we will call these r and t respectively. One can think of ﬁrst identifying by t to form the circle S 1 , and then identifying points on the circle under r. The resulting compact space is a line segment, with 0 ≤ X 25 ≤ πR as a fundamental region. There are ﬁxed points at both ends of the segment, the point πR being ﬁxed by tr. In all, one is identifying the real line under the group tm : tm r :

X 25 X 25

∼ = X 25 + 2πRm , ∼ = 2πRm − X 25 ,

(8.5.3a) (8.5.3b)

for any integer m. Similarly one could identify the k-torus under the reﬂection (8.5.2). In that case there are 2k distinct ﬁxed points, with each of the X m being either 0 or πR. The singular spaces obtained in this way are known as orbifolds. The space obtained by the identiﬁcation (8.5.2) in noncompact space is Rk /Z2 , while the compact space obtained by the same identiﬁcation on the k-torus is T k /Z2 . It is not obvious that string theory on such singular spaces should make sense, but we will see that it does. It is not so diﬀerent from toroidal compactiﬁcation, but it breaks more symmetry (and, later on, supersymmetry) and so will be of interest in constructing string models. The identiﬁcation (8.5.1) or (8.5.2) has two eﬀects. The ﬁrst is that the string wavefunction must be invariant under the reﬂection, equal at identiﬁed points. Second, there is a new sector in the closed string spectrum, in which X 25 (σ 1 + 2π) = −X 25 (σ 1 ) ,

(8.5.4)

257

8.5 Orbifolds

since these are the same point in the identiﬁed spacetime. Strings in this sector are known as twisted states. This is directly parallel to periodic identiﬁcation. There, invariance of the wavefunction made the momentum discrete, while strings that closed only up to the periodic identiﬁcation (winding strings) appeared. One can again make a pictorial argument, analogous to ﬁgure 8.1, to show that twisted strings can be produced from untwisted ones. We will also see below that modular invariance requires them in the spectrum. We will focus on the compact one-dimensional orbifold S 1 /Z2 . Examining ﬁrst the untwisted sector, the spectrum of the periodic theory is reduced by the projection onto invariant states. The eﬀect of r on a general state is ˜ k µ , n, w → (−1) |N, N;

∞ m=1

(Nm25 +N˜ m25 ) |N, N; ˜ k µ , −n, −w ,

(8.5.5)

in particular reversing the compact winding and momentum. We must form linear combinations that are invariant under this. The states that are massless at generic R have n = w = 0, so the projection simply requires that the number of 25-excitations be even. The spacetime graviton, antisymmetric tensor, dilaton, and tachyon survive this projection. The µ α25 modulus α25 −1 ˜ −1 |0; k , 0, 0 also survives — this could have been anticipated, since the radius R can take any value. However, the Kaluza–Klein gauge bosons are no longer in the spectrum. The fate of the extra massless states at the self-dual point is quite interesting, and we will discuss it in detail later. In the sector twisted by r, X 25 is antiperiodic and so has a half-integral mode expansion 25

X (z, ¯z ) = i

1/2 ∞ α

2

α25 α˜25 1 m+1/2 m+1/2 + m+1/2 m+1/2 m + 1/2 ¯ z z m=−∞

.

(8.5.6)

The antiperiodicity forbids any center-of-mass coordinate or momentum, so the string cannot move away from the X 25 = 0 ﬁxed point: to the extent that the oscillations are small, one has X 25 (z, ¯z ) ≈ 0. There are also twisted strings localized at the other ﬁxed point, X 25 (σ 1 + 2π) = 2πR − X 25 (σ 1 ) .

(8.5.7)

The ﬁelds change by tr as one circles the loop. The mode expansion is as above with an additional constant term πR. All other ﬁxed points nπR are images of one of these under t. According to the zero-point mnemonic from section 2.9, the zero-point 1 1 for a periodic boson to 48 for an energy of X 25 is shifted from − 24 1 antiperiodic boson, for a net + 16 . The mass-shell conditions in the twisted

258

8 Toroidal compactiﬁcation and T -duality

sectors are then

4 15 ˜ . m = N− , N=N (8.5.8) α 16 Also, the 25-oscillators make half-integral contributions to the level N. The r-projection again requires the total number of 25-excitations to be even, ˜ In which can also be deduced from the level-matching condition N = N. the twisted sectors the ground states are |T1,2 , the subscript referring to the two ﬁxed points. They are tachyonic, as are the ﬁrst excited states α25 α25 −1/2 ˜ −1/2 |T1,2 . There are no massless twisted states. The extension to the more general case (8.5.2) is straightforward. The quantization is perfectly consistent in spite of the singularities in spacetime. The one complication with orbifolds is that the twisted state vertex operators are not as simple as those for winding states; there is no such explicit formula as the exponential of a free ﬁeld. For two twisted strings and any number of untwisted ones, the tree-level amplitude can be written in operator form T |VU VU . . . |T , which uses only untwisted vertex operators. For four or more twisted vertex operators (the number must be even for the path integral to make sense) two methods have been used. One is the stress tensor method, which we used in section 7.2 to get the partition function on the torus. The other is to go to the covering space on which X 25 is single-valued (this is a genus g surface for 2g + 2 twisted vertex operators on the sphere) and evaluate the path integral. There is a simple result for the tree-level couplings of untwisted states. On the sphere, with all external states untwisted, the twists and projections do not enter into the calculation at all and the amplitudes are the same as in the untwisted theory. For example, the low energy eﬀective action for the massless untwisted ﬁelds is the same as we found for toroidal compactiﬁcation, with the omission of the vector ﬁelds. This property is often useful in analyzing the physics of orbifold theories; it is called the inheritance principle. Consider now the partition function of the X 25 CFT. In the untwisted sector, one is to project the spectrum onto r = +1 states, 1 + r L0 L˜ 0 . (8.5.9) (q¯ q )−1/24 TrU q q¯ 2 The ﬁrst term in the projection operator just gives half the partition function (8.2.9) for toroidal compactiﬁcation. In the term with r in the trace, the diagonal elements of r must have n = w = 0, as one sees from 25 ˜ 25 eq. (8.5.5). The (−1)N +N changes the oscillator sum to 1 − q + q 2 − q 3 + . . . = (1 + q)−1 . The partition function for the untwisted states is then 2

∞ 1 1 |1 + q m |−2 . Ztor (R, τ) + (q¯ q )−1/24 2 2 m=1

(8.5.10)

259

8.5 Orbifolds

The contribution of the twisted sectors is from the sum over half-integer moded oscillations, 1 + r L0 L˜ 0 (q¯ q )1/48 TrT q q¯ 2 = (q¯ q)

1/48

∞ ∞ −2 −2 m−1/2 m−1/2 , 1 − q + 1 + q m=1

(8.5.11)

m=1

the factor of 2 from the number of twisted sectors oﬀsetting the 12 in the projection operator. In all, the partition function can be written as η(τ) η(τ) η(τ) 1 . (8.5.12) + + Zorb (R, τ) = Ztor (R, τ) + 2 ϑ10 (0, τ) ϑ01 (0, τ) ϑ00 (0, τ) The ﬁrst term is already known to be modular-invariant, and it follows from the modular transformation properties given in section 7.2 that the sum of the ﬁnal three is invariant as well. In terms of the path integral on the torus, Ztor (R, τ) is from ﬁelds that are periodic up to translations. The term with ϑab is from the path integral with X 25 (σ 1 + 2π, σ 2 ) = (−1)a+1 X 25 (σ 1 , σ 2 ) , X 25 (σ 1 + 2πτ1 , σ 2 + 2πτ2 ) = (−1)b+1 X 25 (σ 1 , σ 2 ) .

(8.5.13a) (8.5.13b)

For example, the ϑ10 term is from the untwisted sector with r in the trace, that is, from ﬁelds that are periodic in the spatial direction and antiperiodic in the time direction. The ϑ01 term, from the twisted sector with 1 in the trace, has the opposite boundary conditions. These are interchanged by τ → −1/τ. Thus, the twisted sectors are necessary for modular invariance. Twisting Toroidal and orbifold compactiﬁcations are two examples of a useful construction, known as twisting, for constructing new string theories from old. In a given CFT whose symmetries include some discrete group H, we can form a new CFT in two steps. First add twisted sectors, in which the closed string world-sheet ﬁelds are periodic only up to some transformation h ∈ H, φ(σ 1 + 2π) = h · φ(σ 1 ) ,

(8.5.14)

where φ denotes a generic world-sheet ﬁeld. Then, restrict the spectrum to H-invariant states. Although the ﬁelds (8.5.14) are no longer periodic, the vertex operators are periodic because of the projection. So the OPE is still well deﬁned, and it closes because the product of H-invariant operators is H-invariant.

260

8 Toroidal compactiﬁcation and T -duality

In the partition function, we sum over twists h1 in the spatial direction and include a projection operator onto invariant states, 1 hˆ 2 . PH = (8.5.15) order(H) h ∈H 2

The operator hˆ 2 in the trace makes the ﬁelds in the time direction aperiodic by h2 , so in the end we are summing over twists in both the spatial and time directions on the torus, 1 Z= Zh ,h . (8.5.16) order(H) h ,h ∈H 1 2 1

2

The sum over boundary conditions is modular-invariant after a shift of summation variables. The modular transformation τ → −1/τ takes (h1 , h2 ) to (h2 , h−1 1 ), and τ → τ + 1 takes (h1 , h2 ) to (h1 , h1 h2 ). Thus, if we project onto H-invariant states by summing over h2 , we must also sum over the twisted sectors. In more general situations, in particular right–left asymmetric theories, there will be an issue of phases possibly spoiling the naive modular invariance of the sum over boundary conditions, but we will deal with that in chapters 10, 11, and 16. In the case of toroidal compactiﬁcation, the original CFT is the noncompact theory and H consists of tm = exp(2πiRmp) for all integer m. This happens to have inﬁnite order, but we can regulate the sum by putting the system in a large box. For the orbifold, we can think of starting from the toroidal theory and twisting by r, or from the noncompact theory and twisting by r and t. When H is non-Abelian, the path integral boundary conditions are inconsistent for h1 and h2 noncommuting and the path integral vanishes. This also follows from the canonical expression (8.5.15). If φ has spatial twist h1 , then φ (σ 1 ) = h2 · φ(σ 1 )

(8.5.17)

φ (σ 1 + 2π) = h1 · φ (σ 1 ) ,

(8.5.18)

h1 = h2 h1 h−1 2

(8.5.19)

has a diﬀerent spatial twist where and so the diagonal matrix elements of hˆ 2 are zero. We also see that the sectors twisted by h1 and h1 are not independent — they are related by the projection onto h2 -invariant states. Thus the independent twisted sectors are in one-to-one correspondence with the conjugacy classes of H. Twisting can also be thought of as gauging the discrete group H. It may not be obvious that this makes sense, as the discreteness forces the gauge

8.5 Orbifolds

261

parameter to be constant. In particular, there can be no associated gauge ﬁeld. The point is that on a world-sheet with nontrivial closed paths, only gauge-invariant quantities need to be periodic, so the ﬁelds can be aperiodic by an H transformation. Gauging a discrete symmetry adds in these path integral sectors and correspondingly projects onto invariant states. This is just the twisting construction. It is also possible to gauge a continuous world-sheet symmetry to produce a new CFT. This is more involved and will be discussed in chapter 15. The inheritance principle holds for general twisted theories: the path integral with untwisted vertex operators on the sphere is unaﬀected by the twist. c = 1 CFTs We have found two families of compact c = 1 CFTs. These are toroidal compactiﬁcation, restricting the radius to R ≥ α1/2 so as to count only distinct theories, and orbifold compactiﬁcation with the same range of R. These two families are connected. Starting from the S U(2) × S U(2) radius R = α1/2 , twist by r :

X 25 → X 25 + πα1/2 .

(8.5.20)

This produces the toroidal theory at R = α1/2 /2, which by T -duality is the same as at R = 2α1/2 . In terms of the S U(2) currents (8.3.12), r ﬂips the signs of j 1,2 and ˜ 1,2 and so is a rotation by π around the 3-axis in each S U(2). The orbifold r ﬂips the signs of j 2,3 and ˜ 2,3 and is a rotation by π around the 1-axis in each S U(2). But of course these rotations are equivalent under the S U(2) × S U(2), so the resulting theories are the same, the R = 2α1/2 torus and the R = α1/2 orbifold. The corresponding partition functions are equal, Zorb (α1/2 , τ) = Ztor (2α1/2 , τ) ,

(8.5.21)

as can be veriﬁed by theta function identities. This equivalence holds only at these radii. For example, at generic R the toroidal theory has a U(1) × U(1) gauge symmetry while the orbifold has no gauge symmetry. Thus, the two moduli spaces are connected as shown in ﬁgure 8.2. This structure, the moduli space of vacua having distinct branches meeting at special points, will be a very common and important feature in supersymmetric theories. On each branch there is in general a distinct low energy eﬀective ﬁeld theory, here diﬀering in the low energy gauge symmetry. In the neighborhood of the special point there are more light ﬁelds, diﬀerent subsets of which become massive as we move away along the various branches.

262

8 Toroidal compactiﬁcation and T -duality

Rorb α 1/2 D4 4

T

O

I

D3 3

D2 2

Rtor

SU(2) 2 α 1 2 1/2 1/2

Z2 2

Z3 3

Z4 4

Z5 5

Fig. 8.2. Moduli space of c = 1 CFTs. The toroidal branch is horizontal and the orbifold branch is vertical. The special points obtained by twisting by a discrete subgroup of SU(2) are indicated, including the three isolated points obtained from the tetrahedral, octahedral, and icosahedral groups. Other special radii, such as the torus at R = (2α )1/2 , will arise later.

It is instructive to analyze the low energy physics near the meeting of the toroidal and orbifold branches. Describe this point in terms of twisting the S U(2) theory by r , eq. (8.5.20). The massless scalars surviving the twist are j 3˜ 3 and j i˜ j for i, j ∈ {1, 2}. Since these are all untwisted, the potential is the same as before twisting, namely det M. Again the scalar ﬁeld Mij can be diagonalized, by a U(1) × U(1) rotation, and the potential is ﬂat when exactly one diagonal element is nonzero. Now, however, the modulus M33 cannot be rotated into M11 or M22 so these directions are physically distinct. The modulus M33 leaves the U(1) × U(1) unbroken and corresponds to moving along the toroidal branch. The modulus M11 or M22 breaks the gauge symmetry and corresponds to moving along the orbifold branch. Since this modulus is charged, a reversal of its sign is a gauge transformation and so the line of physically distinct orbifolds ends at the point where the branches meet. We cannot simultaneously vary both moduli, because the linear combination is not a ﬂat direction of the potential. That string theory on an orbifold can be equivalent to string theory on a circle is quite remarkable, one more indication that the string does not see spacetime geometry in the same way that we do. Moving along the toroidal branch, there are extra massless states at all multiples of the S U(2) radius, R = kα1/2 . These are the states with

263

8.6 Open strings

(n, w) = (±2k, 0). We can think of these points as obtained, in the T -dual description R = α1/2 /k, by twisting the S U(2) × S U(2) theory by the Zk symmetry 2πα1/2 . (8.5.22) k This multiplies j 1 + ij 2 and ˜ 1 + i˜ 2 by exp(2πi/k) and so the surviving massless scalars are X 25 → X 25 +

j 3˜ 3 ,

j 1˜ 1 + j 2˜ 2 ,

j 1˜ 2 − j 2˜ 1 .

(8.5.23)

The ﬁrst of these changes the radius of the torus and so is a ﬂat direction as always. However, there are no other branches emerging from these points, because the conditions (8.3.23) on the potential for a ﬂat direction are not met in the directions M11 = M22 or M12 = −M21 . The shift (8.5.22) generates a Zk subgroup of S U(2) × S U(2). Let us consider twists by other subgroups. The dihedral group Dk generated by the shift (8.5.22) and the reﬂection X 25 → −X 25 produces the orbifold at radius kα1/2 . The other three discrete subgroups are the tetrahedral, octahedral, and icosahedral groups. These remove all moduli from the CFT, so the twisted CFTs are isolated points, not members of continuous families. Unlike the Zk and Dk twists, which can be deﬁned at all radii, these discrete groups involve elements of S U(2) that exist only at the critical radius and so the radius cannot be varied after twisting. String theory with such a compact space would have only one scalar, the dilaton. These are all of the known c = 1 CFTs, and so give all known 25dimensional bosonic string backgrounds. 8.6

Open strings

The new feature in toroidal compactiﬁcation of the open string is the possibility of a nontrivial Wilson line, a ﬂat background for the gauge ﬁeld. Studying ﬁrst a U(1) gauge theory with some charged ﬁelds, consider the constant background

∂Λ θ iθx25 , (8.6.1) = −iΛ−1 25 , Λ(x25 ) = exp − A25 (x ) = − 2πR ∂x 2πR where θ is a constant. Locally this is pure gauge. The ﬁeld strength vanishes and the ﬁeld equations are trivially satisﬁed. However, the gauge parameter Λ does not satisfy the spacetime periodicity, and so the background has a physical eﬀect. The gauge-invariant quantity that measures this is the Wilson line M

Wq = exp iq

25

dx A25 = exp(−iqθ) .

(8.6.2)

264

8 Toroidal compactiﬁcation and T -duality

Consider ﬁrst a point particle of charge q, with gauge-ﬁxed world-line action 1 ˙M ˙ m2 ˙M . S = dτ (8.6.3) X XM + − iqAM X 2 2

The gauge action is simply −iq dxM AM , so paths that wind around the compact dimension pick up a phase just equal to the Wilson line Wq . The canonical momentum is ∂L qθ p25 = − 25 = v 25 − , (8.6.4) ∂v 2πR ˙ 25 as in eq. (8.4.6). The wavefunctions must be periodic in where v 25 = iX the compact space, so p25 = l/R for integer l and 2πl + qθ . (8.6.5) 2πR The Hamiltonian, which annihilates physical states, is 1 2 H = (pµ pµ + v25 + m2 ) , (8.6.6) 2 so the masses −pµ pµ are shifted due to the dependence of v25 on θ. The same spectrum is readily obtained in a ﬁeld rather than particle description. Note that v25 is just the gauge-invariant momentum −i∂25 − qA25 . One can also make the gauge transformation Λ−1 to set A25 to zero. Charged ﬁelds in this gauge are no longer periodic, picking up a phase exp(iqθ) under x25 → x25 + 2πR. Physical (gauge-invariant) objects are still periodic. The canonical momentum is now shifted, and we have 2πl + qθ v25 = p25 = (8.6.7) 2πR with the same result as before for the gauge-invariant momentum v25 . Let us turn now to the string, and include U(n) Chan–Paton factors. A general constant A25 can be diagonalized by a gauge transformation, 1 diag (θ1 , θ2 , . . . , θn ) . A25 = − (8.6.8) 2πR The gauge ﬁeld is in the diagonal subgroup of U(n), namely U(1)n . The gauge ﬁeld couples to the Chan–Paton factor of a general state as [ AM , λ ], so a string in Chan–Paton state |ij has charge +1 under U(1)i , −1 under U(1)j , and is neutral under the others. Thus it has v25 =

2πl − θj + θi . 2πR The open string spectrum is then v25 =

m2 =

(2πl − θj + θi )2 1 + (N − 1) . 2 2 4π R α

(8.6.9)

(8.6.10)

265

8.6 Open strings

Consider in particular the gauge bosons, where l = 0 and N = 1 and so (θj − θi )2 . (8.6.11) 4π 2 R 2 In generic backgrounds, all the θs are distinct and the only massless vectors are the diagonal ones, i = j. The unbroken gauge group in this case is U(1)n . If r of the θs are equal, the corresponding r × r matrix of vectors is massless, carrying the gauge symmetry U(r). With the n θs equal in sets of ri , the gauge symmetry is m2 =

U(r1 ) × . . . × U(rs ) ,

s

ri = n .

(8.6.12)

i=1

As always, there is a low-energy interpretation. The gauge ﬁeld A25 is a 25dimensional scalar in the adjoint representation of U(n), and its vacuum expectation value breaks the symmetry spontaneously to U(r1 )×. . .×U(rs ). The reader can ﬁll in the details of the eﬀective low-energy action. It is important that with k compact dimensions, the potential contains a term

Tr [Am , An ]2

(8.6.13)

from the ﬁeld strength in the 26-dimensional Yang–Mills action. This forces the gauge ﬁelds in diﬀerent directions to commute in a static solution, so that they are simultaneously diagonalizable. There are then kn moduli from the gauge ﬁeld and k 2 from the metric and antisymmetric tensor. T -duality Now consider the R → 0 limit of the open string spectrum. Open strings with Neumann boundary conditions have no quantum number comparable to w; they can always be unwound from the periodic dimension. So when R → 0 the states with nonzero momentum go to inﬁnite mass, but there is no new continuum of states. The behavior is as in ﬁeld theory: the resulting states move in 25 spacetime dimensions. A seeming paradox arises when one remembers that theories with open strings always have closed strings as well, so that in the R → 0 limit the closed strings move in 26 spacetime dimensions but the open strings only in 25. One can reason out what is happening as follows. The interior of the open string is made of the same ‘stuﬀ’ as the closed string, and so should still be vibrating in 26 dimensions. What distinguishes the open string is its endpoints, so it is these that must be restricted to a 25-dimensional hyperplane. Indeed, recall that one describes the T -dual theory by using the new

266

8 Toroidal compactiﬁcation and T -duality

Fig. 8.3. Open strings with endpoints attached to a hyperplane. The dashed planes are periodically identiﬁed. Two strings are shown, with winding numbers zero and one.

embedding coordinate X 25 (z, ¯z ) = XL25 (z) − XR25 (¯z ) .

(8.6.14)

∂n X 25 = −i∂t X 25 ,

(8.6.15)

Then

with n the normal and t the tangent at the boundary. The Neumann condition on the original coordinate is a Dirichlet condition on the dual coordinate: the X 25 coordinate of each string endpoint is ﬁxed. Let us ﬁrst consider compactiﬁcation without Wilson lines. Then all endpoints are actually constrained to lie on the same hyperplane. To see this, integrate X

25

(π) − X

25

π

(0) = 0

1

dσ ∂1 X

= −2πα v 25

25

= −i

π 0

dσ 1 ∂2 X 25

2πα l =− = −2πlR . R

(8.6.16)

The total change in X 25 between the two ends is an integral multiple of the periodicity 2πR of the dual dimension, so the ends lie on the same hyperplane in the periodic T -dual space. For two diﬀerent open strings, consider the connected world-sheet that results from graviton exchange between them. One can carry out the same argument (8.6.16) on a path connecting any two endpoints, so all endpoints of all strings lie on the

267

8.6 Open strings

0

θ 1R'

θ3R' 2 π R'

θ 2 R'

Fig. 8.4. n = 3 hyperplanes at diﬀerent positions, with various strings attached. Ignoring the tachyon, the lightest strings with both ends on the same hyperplane are massless, while strings stretched between noncoincident hyperplanes are massive due to their tension. When two hyperplanes become coincident, the lightest strings stretched between them become massless.

same hyperplane, as in ﬁgure 8.3. The ends are still free to move in the other 24 spatial dimensions. Now we consider the eﬀect of Wilson lines. Due to the shift (8.6.9) in v25 , the diﬀerence (8.6.16) is replaced by ∆X 25 = X 25 (π) − X 25 (0) = −(2πl − θj + θi )R .

(8.6.17)

In other words, up to an arbitrary additive normalization, the endpoint in state i is at X 25 = θi R = −2πα A25,ii . (8.6.18) Thus there are in general n hyperplanes at diﬀerent positions as depicted in ﬁgure 8.4. The mode expansion in the sector with Chan–Paton state |ij , for an open string winding l times around the compact dimension, is X 25 (z, ¯z ) = θi R − = θi R +

α iR (2πl − θj + θi ) ln(z/¯z ) + i 2π 2

1/2 ∞ m=−∞ m=0

α25 m (z −m − ¯z −m ) m

∞ α25 σ1 m ∆X 25 − (2α )1/2 exp(−mσ 2 ) sin mσ 1 . π m m=−∞ m=0

(8.6.19)

268

8 Toroidal compactiﬁcation and T -duality

The spectrum (8.6.10) becomes 2

m =

∆X 25 2πα

2

+

1 (N − 1) , α

(8.6.20)

where ∆X 25 , given by eq. (8.6.17), is the minimum length for a string in the given sector. Now consider how this picture generalizes if several coordinates are periodic, say X m for p + 1 ≤ m ≤ 25, and we rewrite the periodic dimensions in terms of the dual coordinate. The Neumann conditions on X m again become Dirichlet conditions for the dual coordinates X m , so the open string endpoints are conﬁned to n parallel (p + 1)-dimensional subspaces. The Wilson lines in the various directions become independent coordinates for each subspace. The dual picture is rather odd in that translation invariance is broken by the open string boundary conditions. This simply reﬂects the fact that in the original open string theory winding number is not conserved (a closed winding string can break open and unwind) and winding number is T -dual to momentum. It is important to remember that T -duality is just a diﬀerent description of the same theory, and further that when the compactiﬁcation radii in the original theory are small the T -dual picture is the natural one to use. Figure 8.4 then shows how to interpret the resulting physics. 8.7

D-branes

We have just learned that when an open string theory is compactiﬁed on a small torus, the physics is described by compactiﬁcation on a large torus but with the open string endpoints restricted to lie on subspaces. In fact, these subspaces are new dynamical objects in their own right. Consider the spectrum of massless open strings for the generic conﬁguration in which all θi are distinct, taking for simplicity the case in which only one coordinate is dualized. Ignoring the tachyon, the mass (8.6.20) vanishes only for N = 1 and ∆X 25 = 0. That is, both endpoints are on the same hyperplane, with zero winding. We have therefore the massless states αµ−1 |k; ii ,

V = i∂t X µ ,

α25 −1 |k; ii

V = i∂t X

,

25

(8.7.1a) = ∂n X

25

.

(8.7.1b)

These are of course the same massless states as in the original theory, T -duality simply giving a diﬀerent picture of the same spectrum. For the generic Wilson line, the only massless open string states in the original theory would have been the n massless U(1) vectors. In eq. (8.7.1) we

8.7 D-branes

269

have separated these according to whether the polarization is tangent or perpendicular to the hyperplane. The 25 states with tangent polarization comprise a gauge ﬁeld living on the hyperplane. The state with perpendicular polarization was the gauge ﬁeld in the compact direction of the original theory, and it has a simple and important interpretation in the T -dual theory: it is a collective coordinate for the shape of the hyperplane. This is already evident from the fact that a constant gauge background (8.6.18) corresponds to a uniform translation of the hyperplane. An xµ -dependent background would by the same token go over to an xµ -dependent translation, a curved hyperplane, and the quanta of the ﬁeld A25 ii correspond to small oscillations of the hyperplane. This is the same phenomenon as with spacetime itself. We start with strings in a ﬂat background and discover that a massless closed string state corresponds to ﬂuctuations of the geometry. Here we found ﬁrst a ﬂat hyperplane, and then discovered that a certain open string state corresponds to ﬂuctuations of its shape. We should not be surprised that the hyperplane has become dynamical. String theory contains gravity. A gravitational wave passing through the hyperplane would warp spacetime itself, so the hyperplane could hardly remain rigid. Thus the hyperplane is indeed a dynamical object, a Dirichlet membrane, or D-brane for short. The p-dimensional D-brane, from dualizing 25 − p dimensions, is a Dp-brane. In this terminology, the original U(n) open string theory contains n D25-branes. A D25-brane ﬁlls space, so the string endpoint can be anywhere: it just corresponds to an ordinary Chan–Paton factor. Since T -duality interchanges Neumann and Dirichlet boundary conditions, a further T -duality in a direction tangent to a Dp-brane reduces it to a D(p − 1)-brane, while a T -duality in an orthogonal direction turns it into a D(p + 1)-brane. The case of a nontrivial angle will arise shortly. We could have started by investigating Dirichlet boundary conditions in their own right. The route we have taken, discovering them via T -duality, is a good way to develop many of their properties. It also shows that one can reach a state containing D-branes from an ‘ordinary’ state by a continuous process. That is, take the R → 0 limit of the original theory, leaving n D-branes in a noncompact space. In the superstring, we will use this to argue that D-branes are an essential part of the nonperturbative deﬁnition of the theory. For the bosonic string we will not try to make this argument because there is no strong argument that the theory has a nonperturbative deﬁnition. It is possible that bosonic string theory exists only as a mutilation of superstring theory. It is interesting to look at the U(n) symmetry breaking in the T -dual picture. When no D-branes coincide there is just one massless vector on each, giving gauge group U(1)n in all. If r D-branes coincide, there are new

270

8 Toroidal compactiﬁcation and T -duality

massless states because strings that are stretched between these branes can have vanishing length: the ∆X 25 in the mass formula (8.6.20) vanishes for n = 0 if both i and j are in this set of r. Thus, there are r2 vectors, forming the adjoint of a U(r) gauge group. This is just as found in the dual Wilson line picture. It is striking, however, that there will also be r2 massless scalars from the components normal to the D-brane. We will discuss the signiﬁcance of this below. The D-brane action The massless ﬁelds on the world-volume of a Dp-brane are a U(1) vector plus 25 − p world-brane scalars describing the ﬂuctuations. The low energy eﬀective action for this system is as always a very useful thing to consider. We now take the dual radius R to inﬁnity, focusing on one D-brane in the resulting 26-dimensional spacetime. The world-brane ﬁelds are in interaction with the massless closed string ﬁelds, whose action has already been discussed in section 3.7. Introduce coordinates ξ a , a = 0, . . . p on the brane. The ﬁelds on the brane are the embedding X µ (ξ) and the gauge ﬁeld Aa (ξ). We claim that the action is S p = −Tp

dp+1 ξ e−Φ [− det(Gab + Bab + 2πα Fab )]1/2 ,

(8.7.2)

with Tp a constant to be determined later. Here Gab (ξ) =

∂X µ ∂X ν Gµν (X(ξ)) , ∂ξ a ∂ξ b

Bab (ξ) =

∂X µ ∂X ν Bµν (X(ξ)) ∂ξ a ∂ξ b

(8.7.3)

are the induced metric and antisymmetric tensor on the brane. All features of the action (8.7.2) can be understood from general reasoning. Considering ﬁrst just the spacetime metric and the embedding, the simplest and lowest-derivative coordinate-invariant action is the integral of (− det Gab )1/2 , the world-volume. Note that this term has an implicit dependence on X µ (ξ) through the induced ﬁelds (8.7.3). Expanded around a ﬂat D-brane it gives the action for the ﬂuctuations, just as the Nambu– Goto action describes the ﬂuctuations of a string. The dilaton dependence e−Φ ∝ gc−1 arises because this is an open string tree-level action. The self-interactions of the open string ﬁelds, and their couplings to the closed string ﬁelds, arise ﬁrst from the disk. The dependence on Fab can be understood by using T -duality. Consider a D-brane that is extended in the X 1 - and X 2 -directions, with the other dimensions unspeciﬁed, and let there be a constant gauge ﬁeld F12 on it. Go to the gauge A2 = X 1 F12 . Now take the T -dual along the 2-direction. The Neumann condition in this direction is replaced by a Dirichlet condition, so the D-brane loses a dimension. However, the T -duality relation (8.6.18)

271

8.7 D-branes

between the potential and coordinate implies that the D-brane is tilted in the (1–2)-plane, X 2 = −2πα X 1 F12 .

(8.7.4)

The tilt gives a geometric factor in the action,

dX 1 1 + (∂1 X 2 )2

1/2

=

dX 1 1 + (2πα F12 )2

1/2

.

(8.7.5)

For any D-brane, by boosting to align it with the coordinate axes and then rotating to bring Fab to block-diagonal form, one can reduce the action to a product of factors (8.7.5) in each plane, equivalent to the Fab term in the determinant (8.7.2). This determinant form for the gauge ﬁeld action is known as the Born–Infeld action. It was originally proposed, unsuccessfully, as a solution to the short-distance divergences of quantum electrodynamics. Finally, the dependence on Bab is given by the following argument. The closed string ﬁeld Bµν and the open string ﬁeld Aµ appear in the string world-sheet action as3 i 2 1/2 ab µ ν d σ g ∂ X ∂ X B + i dX µ Aµ . (8.7.6) a µν b 4πα M ∂M Associated with each of these ﬁelds is a spacetime gauge invariance, which must be preserved for the consistency of the spacetime theory. The ordinary gauge transformation δAµ = ∂µ λ

(8.7.7)

is an invariance of the action (8.7.6), the boundary term changing by the integral of a total derivative. The antisymmetric tensor variation δBµν = ∂µ ζν − ∂ν ζµ

(8.7.8)

similarly changes the bulk action by a total derivative, but on a bounded world-sheet this gives rise to a surface term. This can only be canceled if the open string ﬁeld Aµ also transforms under the tensor gauge symmetry, δAµ = −ζµ /2πα .

(8.7.9)

Now, however, only the combination Bµν + 2πα Fµν ≡ 2πα Fµν 3

For the reader familiar with diﬀerential forms, this is i 2πα

B+i M

A. ∂M

The reader can translate the subsequent equations similarly.

(8.7.10)

272

8 Toroidal compactiﬁcation and T -duality

is invariant under both symmetries, and it is this combination that must appear in the action. Thus the form (8.7.2) of the action is fully determined. As a check, it is natural that the combination Gµν + Bµν should appear, since this same combination appears in the world-sheet action in conformal gauge. Incidentally, the action (8.7.2) can also be determined by taking the low energy limit of various open and open plus closed string amplitudes, but this is rather more laborious. For n separated D-branes, the action is n copies of the action for a single D-brane. We have seen, however, that when the D-branes are coincident there are n2 rather than n massless vectors and scalars on the brane, and we would like to write down the eﬀective action governing these. The ﬁelds X µ (ξ) and Aa (ξ) will now be n × n matrices. For the gauge ﬁeld the meaning is obvious — it becomes a non-Abelian U(n) gauge ﬁeld. For the collective coordinate X µ , however, the meaning is mysterious: the collective coordinates for the embedding of n D-branes in spacetime are now enlarged to n × n matrices. This ‘noncommutative geometry’ has proven to play a key role in the dynamics of D-branes, and there are conjectures that it is an important hint about the nature of spacetime. We can gain more insight by considering the eﬀective action. There is now a nonderivative term in the action, a potential for the collective coordinates, which can be deduced by T -duality from a constant Am ﬁeld. For such a vector potential, the ﬁeld strength is just [Am , An ], which becomes (2πα )−2 [Xm , Xn ] in the T -dual picture. Expanding in the ﬁeld strength, the leading term in the action is roughly V ∝ Tr ([Xm , Xn ][X m , X n ]) .

(8.7.11)

This potential has the property that at the point X m = 0 its second derivative vanishes, so that all kn2 scalars are massless; again k = 25 − p is the number of dualized dimensions. However, the space of ﬂat dimensions is smaller. The potential is the sum of squares and so vanishes only if all of the [X m , X n ] vanish. We can then use the U(n) gauge symmetry to go to a gauge in which the X m are all diagonal. There are thus kn ﬂat directions, just the number of diagonal elements. These kn diagonal elements can then be interpreted as the collective coordinates of the n Dp-branes. Thus the potential (8.7.11) correctly interpolates the physics of separated and coincident D-branes. When the X m do commute, the action should reduce to that of n separate D-branes, so we must have S p = −Tp

#

dp+1 ξ Tr e−Φ [− det(Gab + Bab + 2πα Fab )]1/2 $

+O([X m , X n ]2 ) .

(8.7.12)

The determinant is on the world-volume indices ab, while the trace is over

8.7 D-branes

273

the n Chan–Paton indices. The trace is the appropriate U(n) invariant which for diagonal matrices reduces to a sum over the separate D-branes. The full dependence on the commutator is more complicated than the simple potential (8.7.11), with additional couplings to other ﬁelds and corrections at higher orders in the commutator. However, the key property, the form of the ﬂat directions, is unaﬀected. The full dependence can be determined by starting from the Born–Infeld action for the fully Neumann case and performing the T -duality. Incidentally, higher derivative terms involving commutators of ﬁeld strengths cannot be determined by T duality alone. The D-brane tension It is instructive to compute the constant Tp , and for the superstring the precise value will be signiﬁcant. Before we carry out the explicit calculation, it is interesting to note that one gets a recursion relation for Tp from T duality. Note that in a constant dilaton background, the tension of a Dp-brane is Tp e−Φ : this is minus the action per unit volume for a static Dp-brane. Consider now a Dp-brane in which the p directions tangent to the D-brane have been periodically identiﬁed. That is, the Dp-brane is wrapped on a p-torus in spacetime. Its mass is its tension times the volume of the torus, Tp e−Φ

p

(2πRi ) .

(8.7.13)

i=1

Now take the T -dual on one of the periodic dimensions X p . This does not change the mass, being just a new description of the same state. In terms of the dilaton (8.3.31) of the T -dual theory, the mass (8.7.13) is 2πα1/2 Tp e−Φ

p−1

(2πRi ) .

(8.7.14)

i=1

In the dual theory, however, it is a D(p − 1)-brane wrapped on a (p − 1)torus, so its mass is Tp−1 e−Φ

p−1

(2πRi ) .

(8.7.15)

i=1

Equating the masses (8.7.14) and (8.7.15) gives Tp = Tp−1 /2πα1/2 , which determines the Tp up to one overall normalization.

(8.7.16)

274

8 Toroidal compactiﬁcation and T -duality

σ2 σ1

Fig. 8.5. Exchange of a closed string between two D-branes. Equivalently, a vacuum loop of an open string with one end on each D-brane.

To determine the actual value of the D-brane tension we need to calculate a string amplitude. For example, we could deduce it from the gravitational coupling to the D-brane, given by the disk with a graviton vertex operator. However, this involves the yet-unknown ratio gc /go2 , the closed string coupling arising from the vertex operator and the open string coupling from the disk. Instead we can obtain the absolute normalization as follows. Consider two parallel Dp-branes at positions X1m = 0 and X2m = y m . These two objects can feel each other’s presence by exchanging closed strings as shown in ﬁgure 8.5. This string amplitude is an annulus, with no vertex operators, and so can be calculated by the same method as in the previous chapter. The poles from graviton and dilaton exchange then give the coupling Tp of closed string states to the D-brane. Recall that in section 7.4 we calculated the annulus vacuum amplitude in terms of a loop of open strings, but then found closed string poles from the t → 0 limit of the modular integration. Here it is just those poles that we are interested in, but once again the simplest way to calculate the amplitude is to think of it as an open string loop. In fact, the earlier result (7.4.1) carries over with just a few changes. The number of momentum integrations is reduced from 26 to p + 1 and similarly V26 becomes Vp+1 ; the weight hi acquires an additional term y 2 /4π 2 α from the tension of the stretched open string; the Chan–Paton weight n2 becomes 2 (since the string can attach with either orientation). Thus we

275

8.7 D-branes have A = iVp+1 =

∞ dt

t

0

(8π 2 α t)−(p+1)/2 exp(−ty 2 /2πα )η(it)−24

iVp+1 2 (8π α )(p+1)/2

∞

dt t(21−p)/2 exp(−ty 2 /2πα )

0

×[exp(2π/t) + 24 + . . . ] ,

(8.7.17)

where the asymptotics have been obtained just as in section 7.4. The ﬁrst term is from tachyon exchange and so is again an uninteresting bosonic artifact. The integral over the second term gives

24 23 − p A = iVp+1 12 (4π 2 α )11−p π (p−23)/2 Γ |y|p−23 2 2 24π = iVp+1 10 (4π 2 α )11−p G25−p (y) , (8.7.18) 2 where Gd (y) is the massless scalar Green’s function in d dimensions, the inverse of −∇2 . We now need to compare this with a ﬁeld theory calculation of the same amplitude. Since the antisymmetric tensor does not couple to the D-branes, the relevant terms from the spacetime action (3.7.25) are 1 S= 2 2κ

˜ µΦ ˜ − 1 ∇µ Φ ˜∇ ˜ ˜ 1/2 R d X (−G) 6 26

.

(8.7.19)

Recall that the tilde denotes the Einstein metric; this is the convenient form to use because its action decouples from that of the dilaton. The tilded dilaton has been shifted so that its expectation value is zero. In terms of the same variables, the relevant terms from the D-brane action (8.7.2) are p − 11 ˜ ˜ ab )1/2 . Φ (− det G (8.7.20) S p = −τp dp+1 ξ exp 12 We have deﬁned τp = Tp e−Φ0 ; this is the actual physical tension of the Dp-brane when the background value of the dilaton is Φ0 . The ﬁeld theory graph analogous to ﬁgure 8.5 is the exchange of a single graviton or dilaton between the D-branes. To obtain the propagator we ˜ µν − ηµν and to expand the spacetime action to second order in hµν = G ˜ Also, we need to choose a gauge for the gravitational second order in Φ. calculation. The simplest gauge for perturbative calculation is 1 (8.7.21) Fν ≡ ∂µˆ hµν − ∂ν hµˆµ = 0 , 2 where a hat indicates that an index has been raised with the ﬂat space metric η µν . Expanding the action to second order and adding a gauge-

276

8 Toroidal compactiﬁcation and T -duality

ﬁxing term −Fν F νˆ /4κ2 , the spacetime action becomes

1 2 ˜ µˆ ˜ 1 ˆ ˆ Φ . (8.7.22) d26 X ∂µ hνλ ∂µˆ hνˆ λ − ∂µ hνˆν ∂µˆ hλλ + ∂µ Φ∂ 2 8κ 2 3 Inverting the kinetic term, one obtains the momentum space propagators S=−

(D − 2)iκ2 , (8.7.23a) 4k 2 2iκ2 2

hµν hσρ = − 2 ηµσ ηνρ + ηµρ ηνσ − ηµν ησρ . (8.7.23b) k D−2 For later reference we have given these for general D. The D-brane action expanded around a ﬂat conﬁguration is p − 11 ˜ 1 p+1 Φ − haa . (8.7.24) S p = −τp d ξ 12 2 Note that hµν here is traced only over the directions tangent to the D-brane; we have taken ξ to be rectilinear coordinates with metric δab . We can now read oﬀ the Feynman graph, ˜ Φ

˜ =−

Φ

iκ2 τ2p p − 11 A = 2 Vp+1 6 12 k⊥ =

6iκ2 τ2p Vp+1 . 2 k⊥

2

1 1 + 2(p + 1) − (p + 1)2 2 12

(8.7.25)

Comparison with the string result (8.7.18) yields π τ2p = (4π 2 α )11−p . (8.7.26) 256κ2 It is a useful check that, once the dust has cleared, the recursion relation from T -duality is satisﬁed. As one application, consider a state with n 25-branes, which is the same as an ordinary (fully Neumann) n-valued Chan–Paton factor. Expanding the 25-brane action (8.7.12) to second order in the gauge ﬁeld gives τ25 (8.7.27) (2πα )2 Tr(Fµν F µν ) . 4 This relates the open string gauge coupling to the gravitational coupling. Using eqs. (6.5.14), (6.5.16), (6.6.15), and (6.6.18), we can write it as a relation between the vertex operator normalizations gc and go , 4πα go2 go2 = 218 π 25/2 α6 . = (8.7.28) gc κ This has the right form, the square of the open string coupling being proportional to the closed string coupling, but now with the numerical coeﬃcient included. We have essentially obtained this via unitarity, factoring on the closed string poles of the annulus diagram. We could also have

8.8 T -duality of unoriented theories

277

done this more directly, as in exercise 7.9, without the excursion through D-branes. 8.8

T -duality of unoriented theories

The R → 0 limit in an unoriented string theory also leads to interesting new physics. Consider closed strings ﬁrst. To form the unoriented theory we impose Ω = +1 on the states. Following the discussion in section 8.5, we can think about this as gauging Ω. In particular, the transition functions used to build the world-sheet may now include orientation reversal, since it is a gauge symmetry, and this produces unoriented world-sheets. The T -dual theory is obtained by using the coordinate X m (z, ¯z ) = m XL (z) − XRm (¯z ) in place of X m (z, ¯z ) = XLm (z) + XRm (¯z ). In the original description we are gauging world-sheet parity Ω, which acts as Ω : XLM (z) ↔ XRM (z) .

(8.8.1)

In terms of the T -dual coordinates, this is Ω : X m (z, ¯z ) ↔ −X m (¯z , z) , X µ (z, ¯z ) ↔ X µ (¯z , z) ,

(8.8.2a) (8.8.2b)

where again m indexes the coordinates on which the T -dual has been taken and µ those on which it has not. In the dual picture, the symmetry (8.8.2) is thus the product of a world-sheet parity transformation and a spacetime reﬂection. We have seen that gauging world-sheet parity alone produces the unoriented theory, while gauging the reﬂection alone produces an orbifold. The result of the combined projection is known as an orientifold. Orientifolds are rather similar to orbifolds. Separate the string wavefunction into its internal part and its dependence on the center of mass xm , and take the internal wavefunction to be an eigenstate of Ω. The projection onto Ω = +1 then determines the string wavefunction at −xm to be the same as that at xm , up to a sign. For example, the various components of the metric and antisymmetric tensor satisfy Gµν (x ) = Gµν (x) , Bµν (x ) = −Bµν (x) , Gµn (x ) = −Gµn (x) , Bµn (x ) = Bµn (x) , Gmn (x ) = Gmn (x) , Bmn (x ) = −Bmn (x) ,

(8.8.3a) (8.8.3b) (8.8.3c)

where (xµ , xm ) = (xµ , −xm ). There is a minus sign for each m, n and an additional minus sign for each BMN ; the orbifold is the same except that the latter sign would be absent. In other words, the T -dual spacetime is the torus T k identiﬁed under a Z2 reﬂection in the k compact directions, just as in the orbifold construction. In the case of a single periodic dimension,

278

8 Toroidal compactiﬁcation and T -duality

(a)

(b)

Fig. 8.6. Identiﬁcations deﬁning (a) the torus and (b) the Klein bottle.

for example, the dual spacetime is the line segment 0 ≤ x25 ≤ πR , with orientifold ﬁxed planes at the ends. Notice also that away from the orientifold ﬁxed planes, the local physics is that of the oriented string theory. Unlike the original unoriented theory, where the projection removes half the states locally, here it relates the string wavefunction at any point to its value at the image point, as in eq. (8.8.3). One diﬀerence between the orbifold and orientifold constructions is that in the latter there is no direct analog of the twisted states, because the Klein bottle does not have the modular transformation τ → −1/τ. Consider ﬁgure 8.6, which shows the torus and Klein bottle. On the torus, the projection operator inserts a twist in the timelike direction of ﬁgure 8.6(a); rotating the ﬁgure by 90◦ , this becomes a twist in the spacelike direction, implying twisted states in the spectrum. However, if the Klein bottle of ﬁgure 8.6(b) is rotated by 90◦ , the directions of time on the two opposite edges do not match and there is no interpretation in terms of intermediate states in this channel. It should be noted then that orientifold planes cannot be dynamical. Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent ﬂuctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identiﬁcations (8.8.3) become boundary conditions at the ﬁxed plane, such that the incident and reﬂected waves cancel. For the D-brane, the reﬂected wave is higher order in the string coupling. We draw oriented strings with arrows to indicate the orientation. In the unoriented theory, we can either omit the arrow or take a linear combination of the two orientations. The latter picture is more consistent with the idea of gauging the discrete symmetry, and is more general: in an orientifold the parity operation is accompanied by a spacetime transformation, so we cannot just forget the arrow. Although we have introduced the orientifold construction via T -duality, one can consider more general orientifolds that are not simply T -duals

279

8.8 T -duality of unoriented theories

-θ2 R'

-θ1R'

0

θ 1R'

θ 2 R'

π R'

Fig. 8.7. Orientifold planes at 0 and πR , D-branes at θ1 R and θ2 R , and Dbrane images at −θ1 R and −θ2 R . The twist operator Ω acts on any string by a combination of a spacetime reﬂection and reversing the orientation arrow, so that the string running from 2 to the image of 1 becomes a string running from 1 to the image of 2.

of toroidal compactiﬁcations, gauging a group that combines world-sheet parity with various spacetime symmetries. Open strings In the case of open strings, the situation is similar. Let us focus for convenience on a single compact dimension. Again there is one orientifold ﬁxed plane at 0 and another at πR . Introducing S O(n) Chan–Paton factors (the symplectic case works in the same way), a general Wilson line can be brought to diagonal form, and for n even the eigenvalues are paired:

W = diag eiθ1 , e−iθ1 , eiθ2 , e−iθ2 , · · · , eiθn/2 , e−iθn/2 .

(8.8.4)

Thus in the dual picture there are 12 n D-branes on the line segment 0 ≤ X 25 ≤ πR , and 12 n at their image points under the orientifold identiﬁcation. Strings can stretch between D-branes and their images as shown in ﬁgure 8.7. The generic gauge group is U(1)n/2 . As in the oriented case, if r D-branes are coincident there is a U(r) gauge group. However, if now the r D-branes in addition lie at one of the ﬁxed planes, then strings stretching between one of these branes and one of the image branes also become massless and we have the right spectrum of additional states

280

8 Toroidal compactiﬁcation and T -duality

to ﬁll out S O(2r). The maximal S O(n) is restored if all of the branes are coincident at a single orientifold plane. Note that this maximally symmetric case is asymmetric between the two ﬁxed planes. If n is odd, the ﬁnal eigenvalue of W is ±1, so that in the T -dual picture it is ﬁxed to lie on one or the other orientifold plane. Having no image, it is actually a half D-brane, as measured for example by its mass. Also, if we consider for n = 2 the Wilson line diag(1, −1) (which is in O(2) but not S O(2)), then rather than one D-brane and its image there are two half D-branes, one at each ﬁxed plane. The orientifold plane, like the D-brane, couples to the dilaton and metric. The amplitude is just as in the previous section, with the Klein bottle and M¨ obius strip replacing the annulus. In fact, we have already done the relevant calculation in chapter 7. There we found that the total dilaton coupling cancels for S O(213 ). In the T -dual picture, the total dilaton coupling of the orientifold planes cancels that of 212 D-branes (the images don’t count!). If we T -dualize on k = 25 − p dimensions there are 2k ﬁxed planes, where the coordinates are all combinations of 0 and πRm . Thus the eﬀective action for a single ﬁxed plane is 212−k Tp

dp+1 ξ e−Φ (− det Gab )1/2 ,

(8.8.5)

the integral running over the ﬁxed plane. Although the orientifold has no direct analog of twisted sectors, it has been argued that there is a sense in which the twisted states are open strings. We prefer not to emphasize this, as the analogy is not complete: gauging Ω does not of itself introduce world-sheet boundaries. However, there is a deﬁnite physical sense in which it is natural to add open strings to an orientifold theory, to cancel the dilaton tadpole. This cancellation will play an important role in the superstring. Exercises 8.1 (a) Derive the equal time commutator (8.2.21). (b) Work out the details showing that vertex operators of the form (8.2.22) commute at equal time and ﬁnite separation. 8.2 For a single periodic dimension, calculate the scattering amplitude for four closed string tachyons with winding numbers +1, −1, 0, and 0, and vanishing compact momenta. 8.3 We have shown T -duality for spacetime-independent backgrounds. We would expect that a background with a varying metric G25,25 (xµ ) would be T -dual to one with the inverse metric G25,25 (xµ )−1 . In principle we can show this by expanding around the static conﬁguration, but it is useful to show it directly as follows.

Exercises

281

(a) Write the world-sheet action for a string moving in a general massless background, with only the requirement that the background be invariant under translations of X 25 . Let the coordinate X 25 have periodicity R. (b) Show that replacing ∂a X 25 with ∂a X 25 + Aa makes the X 25 translations a local symmetry on the world-sheet. (c) Add a term iφFab to the world-sheet action, with φ(σ) a new ﬁeld and Fab the ﬁeld strength for Aa . The path integral over φ forces the ﬁeld strength to vanish, so the gauge ﬁeld is trivial: this eﬀectively undoes step (b). Argue that in the gauge ∂a Aa = 0 the resulting theory is just a rewriting of the original one. (d) Show that you can reach the gauge X 25 = 0, and then carry out the Gaussian path integration over Aa (treat it classically: ignore the determinant). Show that the resulting action has the expected form for the T -dual theory, if φ is identiﬁed with X 25 times an appropriate constant. To get the transformation (8.3.31) of the dilaton one would need to carry the Aa path integration to one loop. Also, to get the coordinate periodicity of X 25 requires a somewhat careful argument. (References: Buscher (1987, 1988), Roˇcek & Verlinde (1992).) 8.4 Verify the equality of the string partition function for several compact dimensions as calculated directly and via the path integral; see the comment below eq. (8.4.12). 8.5 (a) Show that the spectrum (8.4.26) for a single compact dimension deﬁnes an even self-dual lattice. (b) Do the same for the spectrum (8.4.7). 8.6 In the example of two periodic dimensions, section 8.4, show that the general kinetic term (8.4.35) reduces to the form (8.4.39). Verify the assertions about the action of the various dualities on τ and ρ in the special case that τ and ρ are both imaginary. 8.7 (a) Calculate the scattering amplitude for ripples on a Dp-brane. That is, consider the tree-level amplitude with four collective coordinate vertex operators. Note that the polarizations are orthogonal to all momenta, a major simpliﬁcation. (b) Examine the Regge and hard scattering limits. (c) Take the low momentum limit and compare with what is obtained from the action (8.7.2). 8.8 Repeat the previous problem for two coincident Dp-branes. The collective coordinates now include Chan–Paton factors. Note in particular the change in part (c). 8.9 (a) Calculate the expectation value of n exponential vertex operators in the interior of the disk, with Neumann boundary conditions on p + 1 coordinates and Dirichlet conditions on 25 − p.

282

8 Toroidal compactiﬁcation and T -duality

(b) Extend the calculation to include operators ∂a X M in the interior. 8.10 From the result of the previous exercise, calculate the amplitude for a Dp-brane to emit a graviton. Compare with the Dp-brane eﬀective action and the result (8.7.28) for the couplings. 8.11 (a) For the boundary conditions of exercise 8.9, calculate the disk amplitude with two closed string tachyons: this corresponds to scattering a closed string from a D-brane. (b) Examine the Regge and hard scattering limits. (c) Show that the amplitude has poles from intermediate open strings. Give a physical interpretation. 8.12 (a) Do the calculation of the previous problem for the messier but more interesting case of massless closed strings. (b) Show that there are poles in the graviton amplitude due to the open string collective coordinate, and compare with a calculation using the action (8.7.2). (c) Do the same for the antisymmetric tensor and the open string photon.

9 Higher order amplitudes

In this chapter we discuss higher order amplitudes in string theory. The main goal of the ﬁrst half of the chapter is an understanding of the ﬁniteness and unitarity of string perturbation theory. After discussing the tree-level amplitude in some detail, we consider general amplitudes, using the idea of factorization to relate higher order amplitudes to lower order ones. In the second half, we brieﬂy discuss some ideas that have been put forward to understand all-order and even nonperturbative string theory. The more powerful methods based on supersymmetry will not be developed until volume two. The present chapter is outside the main line of development in this book and can be skipped on a ﬁrst reading. 9.1

General tree-level amplitudes

Chapter 6 was primarily concerned with the calculation of some speciﬁc tree-level amplitudes. It is useful to begin the present chapter with an analysis of the general tree amplitude, as it illustrates some patterns that will arise in the higher order amplitudes. We focus on the closed oriented string, and on the three- and four-point amplitudes, which illustrate the important features. The main issues are convergence and ‘unitarity.’ The last term is used somewhat loosely to cover both unitarity of the S-matrix, S S † = 1, and positivity of the Hilbert space. Of course we already know from chapter 4 that the BRST cohomology has a positive inner product, so what we need to show is that amplitudes respect the BRST invariance: that only BRST-invariant states are produced, and that the amplitudes for BRST-equivalent states are equal. We have given a general argument for the latter in section 5.4, but it is useful to illustrate it explicitly. Before going on to a general analysis let us record some results that are useful in calculation. Suppose we have d inﬁnite ﬂat dimensions and the rest compact (or replaced by a compact unitary CFT), and the tachyon 283

284

9 Higher order amplitudes

vertex operator is normalized gc,d : eik·X : .

(9.1.1)

For example, with toroidal compactiﬁcation of one dimension we would have gc,25 = gc (2πR)−1/2 as in the previous chapter. Then for the matter part of a general OCQ vertex operator, the correct normalization can be ﬁxed by the most singular term in the OPE 2 gc,d + ... . (9.1.2) z 2 ¯z 2 A bar denotes the Euclidean adjoint (6.7.28). The singularity is |z|−4 because the vertex operator is of weight (1, 1). A repetition of the tachyon pole calculation (6.6.7) then determines the normalization

Vj (k; z, ¯z )Vj (k; 0, 0) =

e−2λ ˜c(¯z1 )c(z1 ) ˜c(¯z2 )c(z2 ) ˜c(¯z3 )c(z3 ) : eik·X (z, ¯z ) : =

8πi |z12 z13 z23 |2 (2π)d δ d (k) . 2 α gc,d

(9.1.3)

So all normalization constants in the S-matrix are known in terms of gc,d = κd /2π. Three-point amplitudes Let us ﬁrst express the tree amplitude in several useful forms. The gaugeﬁxed expression (5.3.9) for the string S-matrix reduces, for the general tree-level, three-point amplitude, to a path integral with three ﬁxed vertex operators:

ˆ (∞, ∞)V ˆ (1, 1)V ˆ (0, 0) SS2 (1; 2; 3) = e−2λ V 2 3 1

S2

,

(9.1.4)

where each numeral ‘n’ is an abbreviation for string momentum and state, (kn , jn ). A ˆ is used to designate a vertex operator in the ‘ﬁxed’ picture — that is, not including the b-ghost insertions for the integral over position. As we have discussed in section 5.4, this is the form that comes naturally out of the state–operator mapping, and in the OCQ case takes the form ˆ j = ˜ccVj . (9.1.5) V Using the results from section 6.7 for translating between path integrals on the sphere and matrix elements, this amplitude can also be represented as a matrix element or as an operator product coeﬃcient: ˆ 1 |V ˆ 2 (1, 1)|V ˆ 3 = e−2λ c123 . SS (1; 2; 3) = e−2λ

V (9.1.6) 2

This relates the three-point amplitude in spacetime to a matrix element on the world-sheet, and further to the operator product coeﬃcients which encode the local properties of the CFT.

9.1 General tree-level amplitudes

285

The ﬁrst issue is the decoupling of BRST-null states, so that BRSTˆ 3 in equivalent states have equal amplitudes. Suppose that the state |V the three-point amplitude is null, ˆ 3 = QB |χ . |V (9.1.7) Then ˆ 2 (1, 1)|χ

ˆ 1 | [V ˆ 2 (1, 1), QB ] |χ +

V ˆ 1 |QB V e2λ SS2 (1; 2; 3) =

V =0, (9.1.8) the terms vanishing respectively by the BRST invariance of states 2 and ˆ 2 = {QB , χ} or |V ˆ 1 = QB |χ the amplitude also vanishes. 1. Similarly, if V We have used the Hilbert space formalism here, but it is worthwhile to phrase this same argument in path integral language. Suppose V3 is null, ˆ 3 = QB · X . V (9.1.9) Expand the contour integral on the sphere away from X until it becomes ˆ 1 plus a small loop around V ˆ 2 . These then vanish a small loop around V ˆ ˆ by BRST invariance, QB · V1 = QB · V2 = 0. To this order then, amplitudes are well deﬁned on the physical Hilbert space, the BRST cohomology. The decoupling of the null graviton, associated with spacetime coordinate invariance, is a special case of the decoupling of BRST-null states. It is then natural to wonder whether spacetime coordinate invariance is a piece of some much larger symmetry in string theory. We will discuss some ideas in this direction in section 9.6. We now consider the unitarity of the S-matrix,

∗ Smn Spn = δmp , ?

(9.1.10)

n

which is equivalent to the conservation of probability. It is useful to break up the S-matrix into a term in which nothing happens and a scattering term, Smn = δmn + iTmn .

(9.1.11)

For the scattering matrix T , unitarity implies ∗ Tmp − Tpm =i

∗ Tmn Tpn .

(9.1.12)

n

The T-matrix by deﬁnition involves a total of at least three particles, so for three-particle amplitudes on the sphere the right-hand side of eq. (9.1.12) vanishes. Unitarity then requires TS2 (1; 2; 3) = TS2 (−1; −2; −3)∗ ,

(9.1.13)

where a minus sign is shorthand for replacing an incoming vertex operator by an outgoing vertex operator for the same state, or vice versa. The

286

9 Higher order amplitudes

condition (9.1.13) is satisﬁed if the incoming and outgoing vertex operators are related ˆ −i = −V ˆi. V

(9.1.14)

This is a natural deﬁnition: conjugation takes eik·X into e−ik·X , turning a vertex operator that injects momentum k µ into the world-sheet into one that absorbs it, and similarly for other quantum numbers. The minus sign is included to undo the reversal of the order of ˜cc by the adjoint, so that ˆ −i is obtained simply by conjugating all explicit factors of i in the vertex V operators. Complex conjugation of the path integral then gives

ˆ (∞, ∞)V ˆ 2 (1, 1)V ˆ 3 (0, 0) V 1

∗

ˆ (∞, ∞)V ˆ −2 (1, 1)V ˆ −3 (0, 0) =− V , −1 S2 S2 (9.1.15) implying the unitarity relation (9.1.13). Naively, the path integral is manifestly real and so the relation (9.1.15) should hold without the minus sign. However, we must remember that the path integral has to be deﬁned by speciﬁc analytic continuations, and the explicit result (6.2.6) acquired an overall factor of i in this way. Four-point amplitudes and world-sheet duality The connected four-point amplitude is SS2 (1; 2; 3; 4) = e−2λ

C

ˆ (∞, ∞) V ˆ 2 (1, 1) V ˆ 3 (0, 0) V4 (z, ¯z ) d2 z V 1

S2

.

(9.1.16)

Writing the expectation value as a matrix element gives ˆ 1 |T[ V ˆ 2 (1, 1) V4 (z, ¯z ) ]|V ˆ 3 .

V

(9.1.17)

The ghost expectation value is just a constant for OCQ vertex operators, but to make the connection with the more general case it is useful to move the ghosts around a bit. Using a contour argument one can write the ghost insertion for the d2 z integration as ˆ 4 (z, ¯z ) ] } . V4 (z, ¯z ) = (z¯z )−1 {b0 , [ ˜b0 , V

(9.1.18)

Using this and the condition (4.3.29) that b0 and ˜b0 annihilate the physical states, the matrix element becomes ˜

ˆ 4 (1, 1)|V ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 z L0 −1 ¯z L0 −1 V ˆ 3

θ(1 − |z|)

V ˜ ˆ 2 (1, 1)|V ˆ 1 |V ˆ 4 (1, 1)b0 ˜b0 z −L0 −1 ¯z −L0 −1 V ˆ 3 . (9.1.19) + θ(|z| − 1)

V Let us also write the matrix element after insertion of a complete set of

9.1 General tree-level amplitudes

287

intermediate states, θ(1 − |z|)

i,i

+ θ(|z| − 1)

i,i

2

2

2

2

z α (ki +mi )/4−1 ¯z α (ki +m˜ i )/4−1

ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 4 (1, 1)|V ˆ 3

×

V

z −α (ki +mi )/4−1 ¯z −α (ki +m˜ i )/4−1 2

2

2

2

ˆ 1 |V ˆ 4 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 2 (1, 1)|V ˆ 3 . (9.1.20) ×

V

In this last form the behavior of the integral at z → 0 is evident, and it is the same as for the tachyon amplitudes studied in chapter 6. If ki2 + m2i ˜ 2i are positive for all i, that is, and ki2 + m 4 , (9.1.21) α then the integral converges. Elsewhere it is deﬁned as in eq. (6.6.7), ki2 = (k1 + k2 )2 >

|z|<1

d2 z z α (k

2 +m2 )/4−1

¯z α (k

2 +m ˜ 2 )/4−1

=

δm2 ,m˜ 2 8π . 2 α k + m2 − i

(9.1.22)

One can think of this as analytic continuation from large positive values of m2 , keeping m2 slightly below the real axis. Similarly, the integral as z → ∞ converges for (k1 + k4 )2 > 4/α and is deﬁned elsewhere by analytic continuation. The form (9.1.20) is not useful for studying the z → 1 limit. Instead use the symmetry of the amplitude to interchange vertex operators 2 and 3. The integral for z → 1 is then deﬁned by analytic continuation from the convergent region (k1 + k3 )2 > 4/α . The need to decompose the range of integration and to make possibly diﬀerent analytic continuations for the diﬀerent regions poses no diﬃculty. After the expansion (9.1.20) is integrated as in (9.1.22), the amplitude is expressed as a series of poles in s and u. We know from chapter 6 that the amplitude also has poles in t, but these are not explicit: they come from the divergences of the inﬁnite sums. Similarly the open string amplitude I(s, t) has poles in s and t, but can be written as an inﬁnite sum of terms having poles only in s, or only in t. This property is known as duality, or world-sheet duality to distinguish it from T - and other dualities. We can think of this as arising because the world-sheet can be cut open on diﬀerent curves and so written as sums over intermediate states in diﬀerent ways. Another example is the sum over open string states giving closed string poles on the cylinder. Unitarity of the four-point amplitude Now let us check that four-point amplitudes are equal for equivalent ˆ 3 = QB |χ be null. Using the form (9.1.19), we can commute states. Let |V

288

9 Higher order amplitudes

ˆ 1 |. The BRST operator commutes QB to the left until it annihilates

V ˜ with L0 , L0 and with the vertex operators, but not with b0 and ˜b0 . The nonzero commutators involve

˜ ˜ ˜ b0 ˜b0 z ±L0 −1 ¯z ±L0 −1 , QB = ±b0 ∂¯z z ±L0 −1 ¯z ±L0 ∓ ˜b0 ∂z z ±L0 ¯z ±L0 −1 . (9.1.23) These are total derivatives, illustrating the general principle (5.4.6) that the couplings of null states are proportional to total derivatives on moduli space. We have already seen an example of this for the massless vertex operators in section 6.6. As in that example, the integral vanishes by the canceled propagator argument. The integral at z → 0 is deﬁned by analytic continuation from large positive (k1 + k2 )2 . The surface term vanishes identically there, and so by analyticity must vanish everywhere. The same argument applies to the other limits of moduli space. For the four-particle amplitude on the sphere, the unitarity condition is TS2 (1; 2; 3; 4) − TS2 (−1; −2; −3; −4)∗ =i

j5

= −ie−4λ

dd−1 k5 TS (1; 2; 5)TS2 (−5; 3; 4) 2E5 (2π)d−1 2 + 2 permutations

j5

dd−1 k5 ˆ 1 |V ˆ 2 (1, 1)|V ˆ 5

V ˆ −5 |V ˆ 3 (1, 1)|V ˆ 4

V 2E5 (2π)d−1 + 2 permutations . (9.1.24)

The intermediate particle is on-shell, the sum j5 is over a complete set of physical states, and the right-hand side is summed over all ways to break 1234 into pairs: 12–34, 13–24, and 14–23. Eq. (9.1.24) is what we wish to show. Using the reality condition (9.1.13) in the matrix element (9.1.20), the left-hand side of eq. (9.1.24) vanishes in the region where the z integral converges. The nonzero part comes from the i introduced in the analytic continuation, k2

1 1 − 2 = 2πiδ(k 2 + m2 ) . 2 + m − i k + m2 + i

(9.1.25)

Thus, TS2 (1; 2; 3; 4) − TS2 (−1; −2; −3; −4)∗ 16π 2 i ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 3 (1, 1)|V ˆ 4

= 2λ δm2 ,m˜ 2 δ(ki2 + m2i )

V i i α e i,i + 2 permutations . (9.1.26) The resemblance to the desired unitarity relation (9.1.24) is obvious, and a little bookkeeping and deﬁnition-chasing established the result. The index i in the sum (9.1.26), running over a complete set of string states, labels

9.1 General tree-level amplitudes

289

both internal state and momentum, i ↔ (j, k) .

(9.1.27)

ˆ The delta functions and b0 ˜b0 factor in eq. (9.1.26) project onto the space H ˜ 0 , b0 , and ˜ b0 , which was introduced in the discussion annihilated by L0 , L of the BRST cohomology. Thus, i,i

δm2 ,m˜ 2 δ(ki2 + m2i )b0 ˜b0 |i Gii

i | i

i

= δL0 ,L˜ 0 δ(4L0 /α )b0 ˜b0 =−

iα 16π 2 e2λ

dd−1 k ˆ j (k)| . ˆ j (k)

V |V d−1 2E(2π) ˆ j∈H

(9.1.28)

k 0 =±ωk

The normalization of the basis |Vj (k) follows from the OPE (9.1.2) and the expectation value (9.1.3), which imply 2λ ˆ j (k)|˜c0 c0 |V ˆ j (k ) = 8πie δjj (2π)d δ d (k − k ) .

V (9.1.29) α Using the relation (9.1.28), the form and normalization of eq. (9.1.26) agree with the unitarity condition (9.1.24), leaving one important loose end. The sum in the unitarity relation runs only over the physical Hilbert space, the BRST cohomology, while the ﬁnal sum in (9.1.28) runs over a ˆ We now show that these sums are equal. Consider complete basis for H. ˆ First, take a complete set of ‘unphysical states’ the following basis for H. |a U . That is, no linear combination of these is annihilated by QB . Second, take a complete set of BRST-invariant states |b P which are independent in the sense that no linear combination of them is null. Finally, take the complete set of BRST-null states |a N = QB |a U to complete the basis. Deﬁne the operators P and U:

P|b P = |b P , P|a U = 0 , P|a N = 0 ; U|b P = 0 , U|a U = 0 , U|a N = |a U .

(9.1.30a) (9.1.30b)

Then 1 = P + QB U + UQB .

(9.1.31)

Insert this decomposition into the factorization relation. The ﬁrst term projects onto one state from each cohomology class, the physical spectrum appearing in the unitarity sum. The second and third terms each contain QB , and so vanish by the earlier argument that null states decouple. Thus unitarity is established. It is worth noting that the argument just given is very similar to the argument that amplitudes in gauge theory are simultaneously Lorentz-

290

9 Higher order amplitudes

invariant and unitary. The numerator of the covariant photon propagator η µν , (9.1.32) k2 has a negative eigenvalue, corresponding to a negative norm intermediate state. Choose any null vector nµ such that n · k = 1, and decompose −i

η µν = (η µν − k µ nν − nµ k ν ) + k µ nν + nµ k ν .

(9.1.33)

At k 2 = 0, the ﬁrst term has d − 2 positive eigenvalues and no negative eigenvalues. If the photon is coupled to conserved currents at both ends, the last two terms vanish, so the unitarity relation is satisﬁed with just the d − 2 positive norm states expected for a photon in d dimensions. The decomposition (9.1.31) is the generalization of eq. (9.1.33), with k µ → |a N and nµ → |a U . To conclude, let us review how various general principles are illustrated by this example. BRST null states decouple, but only after integration on moduli space. The potential surface terms from the latter vanish by the canceled propagator argument. The ghost manipulations leading to eq. (9.1.19) have a natural interpretation in terms of the Riemann surface picture from chapter 5. Consider the sphere with vertex operators at 0, 1, z, ∞. To be speciﬁc take |z| > 1 as in the second term of eq. (9.1.19). We can think of this sphere as being formed from two disks, with coordinates z1 and z2 . Put vertex operators at z1 = 0, 1 and at z2 = 0, 1, and join the disks by the identiﬁcation z1 = z/z2 . For the modulus z, we have

z1 ∂z1 = . ∂z z2 z

(9.1.34)

The ghost insertion (5.4.15) for dz is then b0 /z, and similarly for d¯z , as illustrated by eq. (9.1.19). 9.2

Higher genus Riemann surfaces

There are many diﬀerent representations of higher genus Riemann surfaces. The one that will be most useful for our purposes is based on the idea of building higher genus surfaces by sewing together lower genus surfaces. This will be described in the next section, but ﬁrst for completeness we describe some other common constructions. The treatment is necessarily rather brief. Schottky groups Imagine removing two disks from a sphere and joining the edges of the resulting holes. This produces a handle, so the surface obtained in this

9.2 Higher genus Riemann surfaces

291

way is a torus. Repeating g times, with a total of 2g holes, produces a genus g surface. We can make this more precise as follows. Regarding the sphere as the complex z-plane plus point at inﬁnity, consider the mapping z − U z−U =K , (9.2.1) z −V z−V which depends on three complex parameters U, V , and K. This is a M¨ obius transformation, and in particular is holomorphic and one-to-one. The points U and V are ﬁxed under γ. Taking |K| > 1, the surface expands away from U and contracts toward V . A fundamental region for γ is the annular region γ:

|K|

−1/2

z − U < |K|1/2 . < z−V

(9.2.2)

Identifying points under γ glues together the inner and outer boundaries of the annulus (9.2.2), producing a torus. The group generated by g such elements (9.2.1) produces the genus g surface, with restrictions on the parameters so that the sewn circles do not overlap. Every closed oriented Riemann surface can be obtained in this way. The range of the parameters is not simple, in part because some of the modular transformations act in a complicated way. Riemann surfaces with boundary and unoriented Riemann surfaces can be constructed by dividing by one or two reﬂections of a closed oriented surface, as we have illustrated in the lower genus examples. Let us see that the Schottky construction produces the expected number of moduli. Each transformation (9.2.1) has three complex parameters for a total of 3g. One can think of the six real parameters for each handle as being two for the position of each hole, one for the product of the radii of the holes (only the product, because if one hole is made smaller and the other larger we get the same surface), and one for the relative angle at which the holes are sewn. There is redundancy, because an overall M¨ obius transformation of the sphere gives an equivalent surface. Subtracting the six real parameters for this group leaves 6g − 6, the correct number of moduli for genus g ≥ 2 surfaces. For g = 1, two M¨ obius transformations leave the mapping invariant, so two real parameters remain. The Schottky parameterization shows that the moduli space has natural complex coordinates. The path integral has certain holomorphicity properties with respect to these coordinates, though we will not emphasize this. Having constructed the general surface as a coset of the sphere, it is evident that Green’s functions can be constructed by use of the method of images, summing over all images of a source. Functional determinants and other objects of interest can be expressed similarly.

292

9 Higher order amplitudes

B2

B1

A1

A2

B3

A3

Fig. 9.1. Genus-three surface with A- and B-cycles marked.

Fuchsian groups For this construction, consider ﬁrst the world-sheet metric. It is a theorem that by a Weyl transformation one can always make the world-sheet curvature R a constant. At genus zero this gives the round sphere, at genus one the ﬂat torus, and at higher genus a surface of constant negative curvature. The simply connected covering space for any of these will similarly have constant curvature. There are precisely three such simply connected spaces — the round sphere again, the ﬂat plane (the covering space of the torus) and the upper half-plane with metric ds2 =

dzd¯z . (Im z)2

(9.2.3)

The last is the covering space for all the genus g ≥ 2 surfaces. Correspondingly, all genus g ≥ 2 surfaces can be obtained from the upper half-plane by dividing by the action of some discrete group. The elements of this group must be isometries of the metric (9.2.3); these isometries are just the M¨ obius transformations with real coeﬃcients, P S L(2, R). Figure 9.1 shows the genus-three surface with some closed curves drawn in, an A-cycle and B-cycle for each handle. These are nontrivial curves, in that they cannot be contracted to points. If we traverse one of these curves, on the covering space we come back not to the original point but to one of those identiﬁed with it. Thus to each of the cycles is associated an element of P S L(2, R), also denoted Ai or Bi . The A- and B-cycles are complete, in that every closed path is equivalent (homotopic) to a succession of A- and B-cycles. In fact they are slightly overcomplete: the path −1 −1 −1 −1 (A1 B1−1 A−1 1 B1 )(A2 B2 A2 B2 ) . . . (Ag Bg Ag Bg )

(9.2.4)

can be contracted to a point, and the corresponding product must be the identity. Counting moduli, there are three real parameters in each P S L(2, R) element, giving 6g, minus three from the constraint (9.2.4),

9.2 Higher genus Riemann surfaces

293

minus three more because overall conjugation by P S L(2, R) gives an equivalent surface, for 6g − 6 in all. Although the Schottky and Fuchsian parameterizations are similar, they have rather diﬀerent properties. In particular, the complex structure is much less evident in the latter, and the extension to the superstring correspondingly less simple. The period matrix The modulus τ of the torus has a natural g × g matrix generalization at genus g. To see this, we need ﬁrst the Abelian diﬀerentials or holomorphic one-forms. These are world-sheet ﬁelds ωa with the property that in complex coordinates ∂¯z ωz = 0 ,

ω¯z = 0 .

(9.2.5)

There are precisely g such ﬁelds, and g more satisfying the conjugate relation. This follows from the Riemann–Roch theorem (5.3.22). The Abelian diﬀerentials (9.2.5) and their conjugates are just the kernel of the operator P0T , for which the Riemann–Roch theorem is dim ker P0T − dim ker P0 = −χ = 2g − 2 .

(9.2.6)

The operator P0 acts on scalar ﬁelds c and ˜c with h = 0, and so reduces to the ordinary gradient. This always has two zero modes (c = constant or ˜c = constant), and so dim ker P0T = 2g. The line integral of a one-form is a coordinate-invariant quantity. A natural basis for the Abelian diﬀerentials is then ωzi , i = 1, . . . , g, deﬁned by

Ai

dz ωzj = δij .

Then the period matrix

(9.2.7)

τij =

Bi

dz ωzj

(9.2.8)

characterizes the complex structure of the surface. At genus one, where ωz is constant, this is the usual τ. It can be shown that τij is symmetric, so it has 12 g(g + 1) complex parameters. For g = 1, 2, 3 this is equal to the number of complex moduli, but for g ≥ 4 it is greater, so not all period matrices correspond to Riemann surfaces. The theta function (7.2.36) has a natural genus-g generalization, where τ becomes the period matrix and ν, n, a, b become g-component vectors.

294

9 Higher order amplitudes Hyperelliptic surfaces

Here is another description which is sometimes useful. Consider the space S2 × S2 , with coordinates z and y. Consider the two-dimensional submanifold given by the equation y2 =

2k

(z − Zi )

(9.2.9)

i=1

for complex parameters Zi . Projecting down to the z-coordinate, there are two sheets joined across k branch cuts. This produces a surface with g = k − 1 handles. The branch points are singular when projected down to z, but the full surface (9.2.9) is nonsingular there: y is locally a good coordinate. The surface depends on the 2k complex Zi , but three of these are again redundant under a M¨ obius transformation, leaving 2k−3 = 2g−1 complex parameters. For g = 1, 2 this is the same as the number of moduli, but for g ≥ 3 the number of moduli is greater and only special surfaces have the form (9.2.9). These are known as hyperelliptic surfaces. 9.3

Sewing and cutting world-sheets

A useful concept for analyzing string amplitudes is the relation between higher and lower genus amplitudes. A key ingredient is the plumbing ﬁxture, for building up higher genus surfaces from lower ones. Consider two world-sheets, M1 with coordinate z1 in some patch, and M2 with coordinate z2 in some patch. We can join these to form a new worldsheet M as follows. Let q be a complex parameter. Cut out from M1 the disk |z1 | < (1 − )|q|1/2 for some small , and from M2 the disk |z2 | < (1 − )|q|1/2 . Now form a single surface by identifying pairs of points such that z1 z2 = q .

(9.3.1)

This is shown in ﬁgure 9.2. The annular region (1 − )−1 |q|1/2 > |z1 | > (1 − )|q|1/2

(9.3.2)

is sewn as indicated. One notation for this construction is M = M1 ∞M2 (z1 , z2 , q) .

(9.3.3)

The notation emphasizes that the resulting surface depends on the choice of local coordinates. With diﬀerent local coordinates z1 and z2 , one would be cutting out diﬀerent circles and the joined surface will in general be diﬀerent. The two patches might also be in diﬀerent parts of a single

295

9.3 Sewing and cutting world-sheets

M1

M2

M Fig. 9.2. Joining two world-sheets by removing a disk (shaded) from each and gluing the edges as indicated by the dashed lines.

surface M0 . The plumbing ﬁxture construction then adds a handle to the surface. This is denoted M = M0 8(z1 , z2 , q) .

(9.3.4)

The symbols ‘∞’ and ‘8’ are supposed to depict the two sewing processes. The parameter q is not actually needed — taking z1 = z1 q −1/2 , z2 = z2 q −1/2 , and q = 1 gives an equivalent surface. It is useful to introduce q in order to discuss the behavior as a function of q with ﬁxed coordinate patches. We can also reverse this construction. Given any Riemann surface M, and any non-self-intersecting curve C on it, one can always ﬁnd holomorphic coordinates z1 = z2−1 in a neighborhood of C such that C is given by |z1 | = |z2 | = |q|1/2 . Cutting on C (undoing the identiﬁcation z1 z2 = q) and gluing in the disks |z1 | < |q|1/2 and |z2 | < |q|1/2 produces M1 and M2 (or M0 ) such that M would be reconstructed by the sewing operation — either (9.3.3) or (9.3.4) depending on whether the cut along C makes the surface disconnected. For the genus-two example in ﬁgure 9.2, consider the case that q 1. Figure 9.3(a) shows M in this case. We have smoothly interpolated the metric between dz1 d¯z1 and dz2 d¯z2 in the overlap region. The surface

296

9 Higher order amplitudes

(a)

(c)

(b) Fig. 9.3. Three representations of the genus two surface with degenerating cycle.

pinches down to a radius of order |q|1/2 . By a Weyl transformation, one can ﬁnd a metric in which the radius stays of order 1, as shown in ﬁgure 9.3(b). Since a Weyl transformation acts isotropically, the eﬀect is that the length of the handle becomes large, of order − ln |q|. The handle region |z1 | < 1, |z2 | < 1 is equivalent to the cylinder ln |q| < Im w < 0 ,

w∼ = w + 2π ,

(9.3.5)

where z1 = exp(−iw) and z2 = q exp(iw). Figure 9.3(c) shows one more representation of the same surface. The annular region O(|q|) < |z| < O(1) surrounding the small handle is conformally equivalent to the long cylinder of ﬁgure 9.3(b). In terms of ﬁgure 9.3(a), one can think of this picture as a Weyl transformation where all of M2 has been reduced. In all the pictures, it is clear that the surface becomes special in the q → 0 limit. The handle is said to pinch or degenerate. This is an example of a boundary of moduli space. The boundary of moduli space is important, because any divergences must come from there. In fact, we will see that every boundary is of this form, where one or more handles pinches. For the genus-two surface, there is one other way for a single handle to degenerate, if one of the A- or B-cycles pinches. This would correspond to building the genus-two surface from a single torus via the ‘8’ operation (rather than from two tori via the ‘∞’ operation) and taking q → 0. It is a useful fact that by repeated use of the cutting operation, every Riemann surface can be reduced to spheres with three punctures. A puncture is a marked point, with a speciﬁed complex coordinate near

9.3 Sewing and cutting world-sheets

(a)

297

(b)

(c) Fig. 9.4. Some sewing constructions of the genus-two surface with four operators. Each vertex is a sphere with three punctures, each internal line represents the plumbing ﬁxture construction, and each external line represents a vertex operator.

the point; this is the data needed to sew in a plumbing ﬁxture. In other words, every Riemann surface can be constructed from spheres with three punctures via sewing. For example, ﬁgure 9.4 shows three of the many ways to construct the genus-two surface with four vertex operators. There is one complex modulus q for each sewing operation, giving seven in all for each of the constructions in ﬁgure 9.4. These correspond to the 3g − 3 = 3 complex moduli for the genus-two surface plus the positions of the four operators. A graphical decomposition The graphs in ﬁgure 9.4 are reminiscent of Feynman diagrams. We now describe a Feynman-diagram-like decomposition of moduli space. The vertices are rather more complicated than in ﬁgure 9.4, with any number of external lines and with internal corrections from string loops. The point of the construction is to make explicit the nature of the boundary of moduli space. The statement is that the integral over moduli space is given by a sum of Feynman graphs. Each internal propagator in a graph represents the sewing operation, with the sewing parameter q integrated over some standard region such as |q| < 1. The n-point vertex represents a Riemann surface with n punctures. We will see that the surfaces in the vertices must be integrated over some interior region of moduli space, and also summed over genus. The sewing operation refers to a speciﬁc complex

298

9 Higher order amplitudes

Fig. 9.5. Trilinear string vertex, with three marked points and three sewing circles on the sphere.

(a) 1 z4

(c)

(b)

Fig. 9.6. (a) Constructing the sphere with four vertex operators. (b) The region of the z-plane covered by this graph. (c) The additional vertex needed to give the correct amplitude. The four-punctured sphere is integrated over the unshaded region.

coordinate for each point to be sewn, so the deﬁnition of the vertex must include such a choice at each point. The set of genus-g Riemann surfaces appearing in the n-point vertex is denoted Vg,n . External lines represent vertex operators. Let us illustrate this construction. Start with a three-point vertex V0,3 , shown in ﬁgure 9.5 as a sphere with three marked points and three sewing circles. Sewing circles are unit circles in the complex coordinates associated with the respective points. Build a sphere with four points by taking two of these vertices, removing the interior of one sewing circle from each, and sewing them to the ends of a cylinder; this is equivalent to the plumbing ﬁxture construction. This corresponds to the Feynman graph of ﬁgure 9.6(a); the length of the cylinder (with the circumference scaled to

9.3 Sewing and cutting world-sheets

299

2π) is − ln |q|. Integrating over |q| < 1 and summing over the other two channels gives only part of the integration region for the sphere with four points, as shown in ﬁgure 9.6(b). The shaded regions are included but the unshaded interior region is missed. Let us make this more explicit. For the trilinear vertex, put the points on the sphere at the standard 0, 1, ∞ and choose coordinate frames z (0) , z (1) , and z (∞) at each, so the three sewing circles are |z (0,1,∞) | = 1. The choice should be invariant (up to phase) under the six M¨ obius transformations that permute the vertex operators, so that the vertex is symmetric. There are many possible choices, for example az a(z − 1) a , z (1) = , z (∞) = , (9.3.6) z (0) = z−2 z+1 2z − 1 for any constant a. For a ≥ 3, the sewing circles do not overlap, guaranteeing that the plumbing ﬁxture produces a good surface for all q ≤ 1. As the reader can verify, sewing the z (∞) and z (0) patches from two spheres with parameter q produces a sphere with vertex operators at 0, 1, 12 (1 + a2 /q), and 12 (1 − a2 /q). This is equivalent under a M¨ obius transformation to vertex operators at 0, 1, ∞, and z4 = (q − a2 )2 /4a2 q .

(9.3.7)

As q runs over the unit disk, z4 covers the outer shaded region of ﬁgure 9.6(b). Permuting the external vertex operators gives the other two regions. We could try to take a smaller, so the shaded regions grow, but they will eventually overlap and double-count some surfaces while other surfaces will still not be produced. They will never precisely ﬁll the plane, so we need to add in an explicit four-point vertex V0,4 which is given by the sphere with four marked points, integrated over the unshaded part of the z-plane. This does not deﬁne V0,4 completely because there is also a choice of coordinate at each puncture for each value of z4 . It is always possible to make this choice in such a way as to connect smoothly at the boundary between the shaded and unshaded regions. Note that the Feynman graph of ﬁgure 9.6(a) includes all the asymptotic regions z4 → 0, 1, ∞, while the four-point vertex includes only nondegenerate surfaces. The behavior in the limits is then given explicitly by the q → 0 limit of the plumbing ﬁxture: this is why this construction is useful. As another example, sewing two of the circles of a single vertex to the ends of a cylinder produces a torus with one vertex operator. This covers the limit Im τ → ∞ of the moduli space of the torus, but an interior region is missed and must be added by hand as a vertex region V1,1 . This construction can be extended inductively. The sewing operation always increases the parameter r = 3g + n − 2 .

(9.3.8)

300

9 Higher order amplitudes

The vertex V0,3 , the original sphere with three punctures, has r = 1. The 8 operation gives r = r0 + 1, while the ∞ operation gives r = r1 + r2 . Thus, V0,4 and V1,1 both have r = 2. It has been shown that one can construct Vg,n inductively in r, and so cover moduli space as claimed. A more explicit construction is as follows (the proofs can be found in the references). On any Riemann surface there is a unique minimal area metric, deﬁned as the metric of least area subject to the constraint that every nontrivial closed curve (those which cannot be continuously contracted to a point) has length at least 2π. Every point on the surface lies on at least one saturating geodesic, a nontrivial closed curve of length exactly 2π. Saturating geodesics that can be continuously deformed into one another (that is, are homotopically equivalent) cannot intersect. Each set of homotopically equivalent saturating geodesics forms a band (topologically an annulus) on the surface. The diﬀerent bands cover the surface completely, and may overlap. The height h of such a band is deﬁned as the minimum distance between its two bounding geodesics. The idea of a minimal area metric can be extended to surfaces with punctures, with the speciﬁcation that a curve that circles a puncture is regarded as nontrivial. The minimal area metric near a puncture is of the form dzd¯z /z¯z , a semiinﬁnite cylinder of radius 2π. There are thus two kinds of bands: the ﬁnite ones in the interior of the surface, and the semi-inﬁnite ones centered on the punctures, terminating in the interior of the surface. We can now describe Vg,n in this construction. Take all Riemann surfaces with n punctures, with the restriction that the height of any internal band, in the minimal area metric, is less than 2π. Cut oﬀ the semi-inﬁnite cylinders, leaving a stub of length π, and glue in the usual unit disk with vertex operator in the end of the stub. With these vertices, the Feynman graph sum covers moduli space exactly once. It can be shown that the sewing of these minimal area surfaces still produces a minimal area surface. Bands of height less than 2π are explicitly included in the vertices. Bands of height greater than 2π are produced by the sewing operation on the stubs, the resulting height being 2π from the stubs plus − ln |q| from the plumbing ﬁxture; this runs from 2π to ∞ when q is integrated. The minimum length and maximum height restrictions guarantee that degenerating surfaces do not appear in the vertices, so they arise only from one or more qs going to zero, as claimed. 9.4

Sewing and cutting CFTs

We have described higher genus Riemann surfaces as built up by sewing together lower genus surfaces. Using the sewing properties of path integrals, the CFT on the higher genus surface can similarly be related to

301

9.4 Sewing and cutting CFTs

Σi

Ai A

(a)

i

M1

Σi

M2

AiA

(b)

i

M1

M2

Fig. 9.7. (a) The path integral on M is cut open by inserting a complete set of boundary states. (b) Each state is replaced by a disk with local operator.

lower order amplitudes. The relation is shown pictorially in ﬁgure 9.7. For clarity we ﬁrst discuss the case q = 1. The path integral on M is cut open on the circle |z1 | = |z2 | = 1, inserting a complete set of boundary states. Each boundary state is then replaced by a disk with the corresponding local operator, leaving the uncut world-sheets M1 and M2 . The precise form of the result is

. . .1 . . . 2 M =

ij

. . .1 Ai(z1 )

M1

Gij Aj(z2 ) . . .2

M2

.

(9.4.1)

We are allowing arbitrary additional insertions . . .1 on M1 and . . .2 on M2 ; these must of course lie outside the circles |z1 | = 1 and |z2 | = 1 that are being cut. We have not speciﬁed any particular orthonormality for the basis; the appropriate inverse metric Gij is to be determined. The insertions Ai and Aj that replace the handle are implicitly at the origin in the z1 and z2 frames respectively. Now, let us consider the case in which M2 is a sphere, with the sewing coordinate z2 being u and with . . .2 being some local operator Ak at z = 0. We have the identiﬁcations z1 u = 1 from the plumbing ﬁxture and zu = 1 from the sphere, so we can just extend z1 = z through the whole sewn region. The sewn surface M is the same as M1 with an additional

302

9 Higher order amplitudes

operator Ak at z1 = 0. Then eq. (9.4.1) becomes

. . .1 Ak(z1 )

M1

=

1) . . .1 A(z i

M1

ij

Gij Aj(u) Ak(z)

S2

(9.4.2)

.

The two-point function Aj(u) A(z) k S2 is just the inner product

j|k = Gjk . Eq. (9.4.2) then implies Gij Gjk = δ ik

(9.4.3)

Gij = Gij .

(9.4.4)

or

That is, the inner product deﬁned by the two-point function on the sphere is the correct one for sewing. We have thus derived the relation (9.4.1) for the case in which M2 is a sphere with an arbitrary local operator. By the state–operator isomorphism, any boundary state can be obtained in this way. Since the path integral is local, the result depends only on the region of the sewing and not on the rest of the surface, so it holds for arbitrary M2 and . . .2 . Sewing with parameter q is the same as sewing with q = 1 and local coordinates z1 and z2 = z2 /q. We thus obtain eq. (9.4.1) with Aj in the primed frame, and so obtain an extra factor from the conformal transformation back to z2 , (z )

˜

Aj 2 = q hj q¯hj Aj(z2 ) . The result is

. . .1 . . . 2 M =

˜

q hj q¯hj

. . .1 Ai(z1 )

ij

(9.4.5)

M1

Gij Aj(z2 ) . . .2

M2

,

(9.4.6)

where we have taken a basis diagonal in the weights. For the ‘8’ operation the path integrals on M and M0 are related in the same way:

. . . M =

ij

˜

(z2 ) 1) q hj q¯hj Gij . . . A(z i Aj

M0

.

(9.4.7)

It is interesting to consider again the boundary of moduli space, given by the q → 0 limit of one or more handles. We see that the operators of lowest weight will control the behavior. The results (9.4.6) and (9.4.7) resemble the OPE, and in fact (9.4.6) reduces to the OPE when M2 is a sphere and ‘. . .2 ’ is a pair of vertex operators. This analogy is especially clear from ﬁgure 9.3(c), representing the limit as the insertion of a small handle. The sewing relation shows that not only a product of operators, but also a small handle, or any other disturbance that we might sew into a small circle on the world-sheet, is equivalent to a sum of local operators. The result (9.4.6) is an elegant generalization of the OPE. From the point

303

9.4 Sewing and cutting CFTs

of view of M1 , there is a tiny copy of M2 glued in near z1 = 0. This can be expanded in local operators Ai , with coeﬃcient functions Ai . . .2 M2 . From the point of view of M2 , there is a tiny copy of M1 glued in near z2 = 0, which is expanded in local operators Ai , with coeﬃcient functions

. . .1 A i M 1 . We have argued that every Riemann surface can be reduced to thricepunctured spheres. The sewing construction thus relates all amplitudes to the three-point amplitudes on the sphere, which are essentially the operator product coeﬃcients: these are the data that deﬁne the CFT. We will encounter many equivalences between CFTs — we have already seen T -duality, and bosonization (chapter 10) will be another notable example. The sewing construction provides the criteria for two theories to be the same: a one-to-one correspondence between their states (or local operators) and identical OPEs.1 In fact, as we will show in chapter 15, the operator products of the primary ﬁelds alone determine those of all the others. Let us explore this point of view further. One can cut a surface along many diﬀerent sets of curves, so there must be consistency conditions to guarantee that the sewing relation always gives the same result for the expectation values on M. Consider ﬁrst an inﬁnitesimal change in the cutting curve C. A general inﬁnitesimal change leaving M invariant can be obtained by an inﬁnitesimal change in the coordinates used for the sewing, z1

= z1 +

z2 = z2 −

∞ n=−∞ ∞

n z1n+1 ,

(9.4.8a)

−n q −n z2n+1 .

(9.4.8b)

n=−∞

The diﬀerence of the sewn expressions is then proportional to (z )

(z )

Ai 1 Gij Aj 2 − Ai(z1 ) Gij Aj(z2 ) = −

∞

(z2 ) 1 ) ij n Ln ·A(z i G Aj

n=−∞ 2) −Ai(z1 ) Gij L−n ·A(z j

+ (anti) , (9.4.9)

where we have used the conformal transformation (2.9.6), and ‘anti’ de˜ n . The vanishing of the notes the same form with n → ∗n and Ln → L quantity in braces can be shown by an argument parallel to that used to derive eq. (9.4.4). This is just the statement that one can pull the contour 1

We should note that when there is more than one (2, 0) plus (0, 2) ﬁeld that could play the role of Tab , as in the free-ﬁeld examples of section 2.5, it is necessary to specify which is the energy-momentum tensor. This choice determines the conformal transformations of operators, which enter implicitly into the sewing relation.

304

9 Higher order amplitudes

integral that deﬁnes the generator through the sewn handle; the change from Ln to L−n is from an inversion of coordinates. All other contour integrals for conserved charges can be pulled through as well. There are also sets of curves that are not related by a continuous deformation. The four-point amplitude on the sphere can be written in terms of three-point functions by cutting on curves circling diﬀerent pairs of operators. Equality between diﬀerent sewings is the associativity of the OPE, ﬁgure 2.8. Another example is given by the torus: we can reduce the torus to the sphere by cutting on the A-cycle, or the B-cycle, or any other nonself-intersecting closed curve. In terms of the plane with identiﬁcations, a closed curve C that begins at (σ 1 , σ 2 ) = (0, 0) must end at an identiﬁed point 2π(m, n) .

(9.4.10)

All curves with the same m and n values can be continuously deformed into one another, so we need to consider the relation between sewing on curves of diﬀerent m and n. One can show that for a nontrivial non-selfintersecting curve, m and n must be relatively prime, which implies that there are integers p and q such that mp − nq = 1. There is then a modular transformation (σ 1 , σ 2 ) = (σ 1 , σ 2 )

m n q p

,

(9.4.11)

such that the curve C comes back to (σ 1 , σ 2 ) = 2π(1, 0) ,

(9.4.12)

which is the A-cycle. So cutting on a general nontrivial non-self-intersecting curve is equivalent to cutting on the A-cycle for a modular-equivalent value of τ. The consistency of the sewing construction is thus equivalent to modular invariance. Every set of sewing curves may be transmuted into any other by a series of two moves: the four-point associativity transformation on the sphere and the modular transformation on the torus with a single vertex operator. Figure 9.4 makes this plausible. For example, ﬁgures 9.4(a) and 9.4(b) are related by one associativity condition. Repeated applications of associativity can bring the sewing to the form of ﬁgure 9.4(c), where all loops are expressed in terms of the one-loop, one-point function. So these are the conditions for an arbitrary set of operator product coeﬃcients to deﬁne a sensible conformal ﬁeld theory. The program of solving the duality constraints is known as the conformal bootstrap. It has been carried out only in cases where additional restrictions have been introduced, as we will discuss in chapter 15.

305

9.5 General amplitudes

All the considerations of this section extend to world-sheets with boundary, where in addition to degenerating handles there may be degenerating strips. The consistency conditions have been generalized to this case. 9.5

General amplitudes

We now have assembled all the ingredients that we need in order to understand the consistency of general string amplitudes. Having given a graphical decomposition of moduli space, the sewing formalism allows us to give a diagrammatic expression for the general string amplitude. Consider ﬁrst the propagator, from the plumbing ﬁxture. The ghost insertion for the modulus q is

dz ∂z1 b0 = bz1 z1 , 2πi ∂q z2 q

(9.5.1)

q . Including these insertions, and using the and similarly for q¯ it is ˜b0 /¯ sewing expression (9.4.6), the factors associated with the propagator are 8πδm2 ,m˜ 2 Gij b0 ˜b0 d2 q α (k2 +m2 )/4 α (k2 +m˜ 2 )/4 ij ˜ i i i i . (9.5.2) G b0 b0 = 2 i i 2 q¯ q q α (ki + mi − i) |q|<1 q¯ The n-string coupling is given by inserting vertex operators into the surfaces appearing in the vertices of the sewing construction, g(i1 , . . . , in ) =

g

m

d t

Vg,n

m

k=1

Bk

n l=1

ˆi V l

.

(9.5.3)

g

where m = 6g + 2n − 6 is the number of moduli. The Feynman graphs constructed from the couplings (9.5.3) and propagators (9.5.2) then reproduce the full string amplitude. The label i again stands for the state and momentum of each string. The tree amplitude in section 9.1 illustrates this structure. The ﬁrst issue is the ﬁniteness of the amplitude. We assert that divergences can arise only from the boundary of moduli space, as in the various low order examples. At ﬁnite values of the moduli, the sewing formula gives a sum over an inﬁnite number of string states, but this sum converges because it is ultimately just the expansion of a quantum state in a complete set. Actually, convergence is guaranteed only in a unitary CFT, but the nonunitary parts of the string CFT are quite tame. The X 0 CFT is deﬁned by analytic continuation from a unitary CFT, while the ghost CFT just generates a Faddeev–Popov determinant and so is ﬁnite if we have made a good choice of gauge. Now, the boundary has been made explicit as the q → 0 limit of the various propagators, and the integral (9.5.2) is one that we have encountered several times before. It is

306

9 Higher order amplitudes

deﬁned via analytic continuation, giving an ordinary ﬁeld-theoretic propagator. The only singularities are then the poles in propagators, which represent production of on-shell particles in the intermediate states. In particular they are long-, not short-, distance divergences. The other issue is unitarity. We studied this for the four-point tree amplitude in some detail. Having brought the general amplitude to a Feynman-diagram form, rather than work out the combinatorics in detail we will simply cite standard ﬁeld theory results on the unitarity of the diagrammatic expansion. Also, the demonstration at the end of section 9.1 that the sum over intermediate states reduces to a sum over physical states generalizes as in ﬁeld theory. In summary, the issues involved in the perturbative ﬁniteness and unitarity of string theory appear to be well understood. Observe once again that we have not assumed 26 ﬂat dimensions, but merely d ﬂat dimensions plus a compact positive norm CFT of central charge 26 − d. Of course, any discussion of the unitarity of the ﬂat spacetime S-matrix for the bosonic string is somewhat formal, since ﬂat spacetime is not a solution of the theory — ﬁrst there is the tachyon instability, then the cosmological constant induced by loops. An alternative approach to the ﬁniteness and unitarity of string theory, based on the light-cone gauge, has also been extensively developed. This gauge gives up covariance, but has certain advantages — it makes possible a rather explicit description of moduli space, and the amplitudes can be represented in a ﬁeld-theoretic form with a Hamiltonian that is local in the light-cone time x+ . Covariance has then been established by showing that the perturbation theory is equivalent to the covariant perturbation theory. String divergences Before moving on, let us expand somewhat further this point that string divergences are long-distance eﬀects. Consider the one-loop amplitude with n vertex operators, taking the limit that m operators come together with τ held ﬁxed as in ﬁgure 9.8(a). The string loop of course includes gravity, and ﬁgure 9.8(a) is reminiscent of the graviton loop of ﬁgure 1.1(c), which was UV divergent when the interaction vertices came together in spacetime. There is indeed a divergence in this string amplitude, but it is from long spacetime distances, not short ones. Figure 9.8(b) shows a conformally equivalent picture where the m vertex operators are connected to the rest of the torus by a long cylinder. Cutting the cylinder open and inserting a complete set of states, we obtain the usual expression (9.5.2), with q → 0 being the limit where the vertex operators become coincident. So the only singularities are the usual poles from propagation over long spacetime

9.5 General amplitudes

(a)

307

(c)

(b) Fig. 9.8. (a) Torus with n vertex operators, in the limit that m of the operators come together. (b) Conformally equivalent picture: the m vertex operators at the end of a long cylinder. (c) Sewing construction, in terms of the product of a sphere with m + 1 vertex operators and a torus with n − m + 1.

distances. Conformal invariance once again changes short distances on the world-sheet into long distances in spacetime. The cases m = n and m = n − 1 present special problems. When m = n, meaning that all the vertex operators are coming together, momentum conservation forces k µ to vanish. For massless states, the pole is then 1/0 and the usual analytic continuation does not help because momentum conservation forces the amplitude to sit right on the massless pole. This is a tadpole, just as in section 7.4, and the resolution is the same, a correction to the background ﬁelds. The divergence here is proportional to the torus amplitude with one vertex operator. In the other problematic case, m = n − 1, all the vertex operators but one, say Vn , come together. Momentum conservation requires that the momentum ﬂowing through the cylinder be k µ = −knµ . However, conformal invariance requires that kn2 = −m2n , so the intermediate momentum is ﬁxed on top of a pole for those intermediate states with m2j = m2n . Once again, there is an analogous divergence in ﬁeld theory, shown in ﬁgure 9.9: the internal propagator is 1/0. This divergence occurs whenever the mass of the internal line j is the same as that of the on-shell external particle n. The interpretation here is simple. The loop represents a radiative correction to the mass of the particle n. The S-matrix is deﬁned with the external momentum taken on the true mass shell, including radiative corrections, and it is again free of this IR divergence. Note that the issue here is not some UV divergence internal to the loop, but the IR divergence on the

308

9 Higher order amplitudes

* Fig. 9.9. Field-theory graph analogous to the case m = n − 1. The kinematics force the starred propagator to diverge. The interpretation is a (ﬁnite) mass shift due to the loop.

starred propagator which appears even when the mass correction is ﬁnite. The resolution in the string case is the same. The momentum appearing in the vertex operator must be at the true physical mass; this is not in general the same as the tree-level mass, the one whose square was always a multiple of 1/α . Note that for certain massless ﬁelds — in particular gauge bosons — gauge invariance guarantees that they will remain massless to all orders. Incidentally, for massive states the mass shift is usually complex, because these states are unstable to decay into lighter string states. There is a puzzle here. We saw early on that consistency of string theory, the condition that the world-sheet theory be conformally invariant, required the beta functions (3.7.14) to vanish. These beta functions contain no information about higher genus surfaces — they correspond to the treelevel string ﬁeld equations. What happens when we try to expand around a loop-corrected background? In fact things work out just right. The dangerous limit in ﬁgure 9.8 gives rise not only to a divergence, but also to a conformal anomaly. To see this heuristically, picture the limit as in ﬁgure 9.3(c), as a small handle glued into the sphere. The small handle can be represented by a series of local operators, and in particular from the massless level there is q ¯ µ dqd¯ . (9.5.4) ∂X µ ∂X q¯ q Now, cut oﬀ the divergence in a coordinate-invariant way by requiring the size of the handle to be greater than a; that is, |q|eω > a, where the metric on the sphere is dzd¯z e2ω . The lower limit on the integral (9.5.4) gives ¯ µ ln(ae−ω ) . (9.5.5) − 4π∂X µ ∂X Thus we have new dependence on the Weyl factor of the metric: this G , eq. (3.7.14). To obtain a Weylcorrects the sigma-model Weyl anomaly βµν invariant theory, the background ﬁelds must satisfy the full string-loopcorrected Weyl invariance conditions (or, equivalently, BRST invariance). The Weyl anomaly from the shifted background cancels that from the small handle. This is the Fischler–Susskind mechanism. From ﬁgure 9.3(c),

9.5 General amplitudes

309

we can think of the divergence and the Weyl anomaly as world-sheet UV eﬀects, not from the usual short-wavelength ﬂuctuations of the ﬁelds, but from a small topological ﬂuctuation. We should emphasize that on a ﬁxed Riemann surface of any genus, the local structure is unaﬀected by the topology; it is the integration over Riemann surfaces that produces the divergence and Weyl anomaly. This can also be regarded as a breakdown of the canceled propagator argument. Because the kinematics forces the amplitude to be at the pole, we cannot appeal to analytic continuation to eliminate possible surface terms under a BRST variation. In fact they appear, corresponding to the Weyl anomaly (9.5.5). In the loop-corrected background, the BRST anomaly from the shift of the background cancels that from the boundary of moduli space — the stringy technicalities all ﬁt together to give a consistent spacetime picture. To implement this in string perturbation theory is rather clumsy, but fortunately it is rarely necessary. First, in most supersymmetric theories the backgrounds are uncorrected. Second, for the analysis of long-distance physics in string theory it is usually most eﬃcient to proceed in two steps, ﬁrst using string theory to derive the form of the low energy eﬀective action and then using the eﬀective ﬁeld theory at long distance. One then does not need to deal with the IR divergences at the string level. In the present case, the tadpoles imply a potential energy term δS = −

Λχ

dd x (−G)1/2 e−χΦ

(9.5.6)

χ

in the eﬀective action. Here Λχ is the tadpole from Euler number χ, and the dependence of the action on Φ follows from the fact that eΦ is the string coupling. The separate tadpoles for the graviton and dilaton can be extracted from the string amplitude, with some eﬀort. The same idea extends to the vertex operator correction from the mass shift, and to other IR divergences such as those of exclusive amplitudes in theories with massless particles. These ﬁeld-theory divergences must also appear in the corresponding string-theory amplitudes, since string theory reduces to ﬁeld theory at long distance. Again we emphasize that while UV divergences in ﬁeld theory generally signify a fundamental limitation on the range of validity of the theory, IR divergences usually mean that one is simply asking the wrong question (wrong background, wrong mass, exclusive versus inclusive cross-section, etc.) in what is otherwise a perfectly consistent theory. Any such divergences in string theory should then be dealt with precisely as in ﬁeld theory. Let us review what we have learned. Modulo problems particular to the bosonic string, the S-matrix is ﬁnite and unitary. The basic mechanism is modular invariance, which cuts oﬀ the UV divergences without spoiling

310

9 Higher order amplitudes

the spacetime gauge invariance. Of course, knowing that string theory makes sense in weak coupling perturbation theory around ﬂat spacetime is hardly enough. We would like to know that the theory makes sense beyond perturbation theory, and have some means of calculating nonperturbative corrections. Also we would like to know what string theory implies for the nature of spacetime at short distance. In the remainder of this chapter we brieﬂy describe some of the attempts that have been made in these directions. 9.6

String ﬁeld theory

It is interesting to compare the approach we have taken in string theory to the way quantum ﬁeld theory is formulated. The world-sheet path integral is analogous to a sum over particle paths, often called the ﬁrstquantized description. In quantum ﬁeld theory, on the other hand, we start with a path integral (or its Hamiltonian equivalent) over spacetime ﬁelds. One could instead develop quantum ﬁeld perturbation theory in the ﬁrst-quantized form; it is necessary to introduce explicit interaction vertices, spacetime points where n particle paths end. In string theory this second step is not necessary — there are no special points, and the sum over topologies produces the interactions. However, the result is only the perturbation expansion, and there are important phenomena that are not contained in this expansion. In the Standard Model, these include quark conﬁnement and the ﬁnite-temperature baryon number violation of the weak interaction. There does not seem to be any useful sense in which these are contained in a resummation of the perturbation series; rather they require a more general formulation of the theory. In the previous section we have expressed string amplitudes in a propagator and vertex form that looks like the perturbation theory for an inﬁnite-component ﬁeld theory. Could we instead derive this from a path integral over an inﬁnite set of spacetime ﬁelds? And is there some principle, extending gauge invariance and general coordinate invariance, that determines the form of the theory? This is the subject of string ﬁeld theory. We will be able to sketch here only a few of the ideas, ones that are particularly close to the approach that we have been taking. The reader should consult the literature for more details. We will start with the guess that we should take the string wavefunctional Ψ[X] and elevate it to a ﬁeld operator, or equivalently to a path integral variable of integration — so-called second quantization. Now, Ψ[X] depends on the curve X µ (σ) traced out by the string in spacetime, and roughly speaking creates or destroys a string in this conﬁguration. However, we need to be more precise. For example, should we require Ψ

9.6 String ﬁeld theory

311

to be invariant under reparameterization of the curve, and should it also depend on the Polyakov metric gab ? Things seem to work much more nicely if one starts with the BRST-invariant form of the theory, including the Faddeev–Popov ghosts, and then second quantizes. From the commutation relations, b and c are conjugate to one another (as are ˜b and ˜c), so if for example we treat c and ˜c as coordinates the wavefunctional is Ψ[X, c, ˜c ]. We then have a path integral running over all such functionals,

[dΨ] exp(iS [Ψ]) .

(9.6.1)

We will consider ﬁrst the open string, describing one particular approach. What symmetry principle will restrict the string action? From our study of the string vertex operators and null states, we might guess the symmetry δΨ = QB Λ

(9.6.2)

for any functional Λ. There is an extremely simple action with this invariance, S0 =

1

Ψ|QB |Ψ . 2

(9.6.3)

The equation of motion is just QB Ψ = 0 .

(9.6.4)

One way to see that this is reasonable is by expanding the string ﬁeld Ψ in terms of an inﬁnite number of ordinary ﬁelds, one for each internal state of the string. Let Ψi [X , c, ˜c ] be a complete set of wavefunctions for the internal modes — all but the center-of-mass xµ . Then a general state can be expanded: Ψ[X, c, ˜c ] =

Φi (x)Ψi [X , c, ˜c ] ,

(9.6.5)

i

the expansion coeﬃcients being functions of the remaining variables xµ . The string ﬁeld path integral becomes an inﬁnite product of path integrals over the component functions,

[dΨ] →

[dΦi ] .

(9.6.6)

i

As usual, we can start with the ground state functional Ψ0 and build the remaining states with the raising operators. For example, keeping all

312

9 Higher order amplitudes

(a)

(b) 1

2

Fig. 9.10. (a) Ψ1 ∗ Ψ2 , convoluting the right half of Ψ1 with the left half of Ψ2 . One can think of this as deﬁned by a path integral over the shaded region, whose width is then taken to zero. (b) Ψ, convoluting the right and left halves of Ψ.

states at the ﬁrst two levels,

Ψ = ϕ(x) + Aµ (x)αµ−1 + B(x)b−1 + C(x)c−1 + ϕ (x)c0

+ Aµ (x)αµ−1 c0 + B (x)b−1 c0 + C (x)c−1 c0 + . . . Ψ0 . (9.6.7) Here, ϕ is the tachyon ﬁeld and Aµ the gauge ﬁeld, while B and C are the Faddeev–Popov ghost ﬁelds associated with the spacetime gauge symmetry. The primed ﬁelds can all be gauged away or are auxiliary (that is, they satisfy algebraic rather than diﬀerential equations and can be eliminated). One can then check that the symmetry (9.6.2) includes the ordinary gauge transformation on Aµ , while the action is of Klein–Gordon form for ϕ and of Maxwell form for Aµ . The BRST string ﬁeld formalism thus gives a very compact generalization of spacetime gauge invariance. Interactions also work out rather nicely. Deﬁne a product operation Ψ1 ∗Ψ2 on string ﬁelds by the functional integral depicted in ﬁgure 9.10(a), forming the overlapof the right half of Ψ1 with the left half of Ψ2 . Deﬁne also an operation , giving the overlap of the right and left halves of the single string wavefunctional as shown in ﬁgure 9.10(b). To make an interacting theory, add a nonlinear term to the gauge transformation, δΨ = QB Λ + gΨ ∗ Λ − gΛ ∗ Ψ .

(9.6.8)

The ∗ product is associative, and one can show by a contour argument that QB acts as a derivation, QB (Ψ1 ∗ Ψ2 ) = (QB Ψ1 ) ∗ Ψ2 + Ψ1 ∗ (QB Ψ2 ) .

(9.6.9)

From these it follows that the symmetry (9.6.8) closes. Using the cyclic property,

Ψ1 ∗ Ψ2 =

Ψ2 ∗ Ψ1 ,

(9.6.10)

313

9.6 String ﬁeld theory VΨ

VΨ

(a)

VΨ QB

(b)

VΨ

VΨ

Fig. 9.11. CFT representation of the string ﬁeld action: (a) the kinetic term; (b) the interaction. A z 2/3 conformal transformation is used to turn the usual semi-circle around each vertex operator into a wedge.

we can write the invariant action 1 2g S= Ψ ∗ QB Ψ + Ψ∗Ψ∗Ψ . (9.6.11) 2 3 This is also usefully written in terms of CFT, replacing Ψ with a half-disk with corresponding vertex operator VΨ , 1 2g

VΨ QB ·VΨ D2 +

V Ψ V Ψ V Ψ D 2 . (9.6.12) 2 3 The vertex operator positions and conformal frames are not explicitly denoted in eq. (9.6.12) but are as shown in ﬁgure 9.11. A z 2/3 mapping is used to ﬁt the three vertex operator half-disks into the disk. This action passes an important check. The path integral on the disk is nonzero only for insertions of total Qg = 3. The vertex operators for the physical states all have ghost number 1, so both terms in the action (9.6.12) are nonvanishing. Since the action (9.6.11) is invariant, we might expect that for the gauge ﬁeld it reduces to the non-Abelian Yang–Mills action, and this is the case. The Siegel gauge is b0 Ψ = 0. In the kinetic action, only the term c0 L0 from QB survives. Since in components L0 ∝ p2 + m2 , this is analogous to the familiar covariant gauge for electrodynamics. The propagator S=

∞ 0

dt exp(−tL0 ) = L−1 0

(9.6.13)

is simply the integral over strips of any length. The vertex is ﬁgure 9.11(b) with the wedges removed, joining the ends of three strips in a ‘Y.’ This has been shown to be equivalent to open string theory described as a sum over Riemann surfaces, including in particular a complete single cover of moduli space. On the one hand this is not surprising: the string ﬁeld theory has the spacetime gauge invariance (9.6.8), so that null states will

314

9 Higher order amplitudes

decouple from the S-matrix. We have argued that the decoupling of null states determines the form of the string amplitudes including the need for a complete cover of moduli space (else total derivative terms will give null state couplings). On the other hand moduli space is complicated and few explicit descriptions are known. The one produced by this string ﬁeld theory, in terms of a sum over Feynman graphs and an integral over strip lengths for each, was discovered by mathematicians only shortly before its construction through string ﬁeld theory.2 Let us point out a strange feature of this ﬁeld theory. We have stated that the string ﬁeld theory covers the entire moduli space, but we know from the annulus that this includes limits that produce closed string poles, even though there are no explicit closed string ﬁelds. The closed string poles come from strips going to zero, rather than inﬁnite, length.3 The precise statement is that the open string ﬁeld path integral (9.6.1) generates the amplitudes of open plus closed string theory (except that surfaces with no boundaries at all, to which open strings cannot couple, are not generated). In this respect it is as though closed strings were in some imprecise sense ‘bound states’ of open strings. For a theory with only closed strings we can try to repeat the above steps with a closed string ﬁeld, though it is somewhat more complicated. Already for the free action (9.6.3) the ghost number does not work out: we need an extra c ghost to make the matrix element between physical states nonvanishing. There is a remedy, which is to require the string ﬁeld Ψ and the gauge transformation Λ to be annihilated by b0 − ˜b0 and by ˜ 0 . The action L0 − L S0 =

1

Ψ|(c0 − ˜c0 )QB |Ψ

2

(9.6.14)

is then invariant under δΨ = QB Λ. Going on to the interacting theory there is no associative product analogous to ∗, and correspondingly no simple cover of moduli space. As in the previous section we need to add in higher vertices Vg,n , with more powers of the ﬁeld and also with internal handles, in order to cover moduli space. The resulting action, like the open string action, can be obtained from a symmetry principle using what is known as the Batalin–Vilkovisky formalism. Gauge ﬁxing then leads to the perturbation theory described in the previous section. 2

3

Another physically motivated description of moduli space comes from the light-cone gauge, where the moduli are the interaction times and the momenta p+ of the intermediate states, and also a twist angle for each intermediate closed string. For this reason, this form of string ﬁeld theory would not be suitable for our purposes of the previous section. We would need to add stubs to the vertex to eliminate the unexpected divergences, and then to add in closed string propagators and vertices to cover the omitted region of moduli space.

315

9.7 Large order behavior

The derivation of string theory from spacetime gauge symmetry is extremely elegant, particularly for the open string. The inclusion of closed strings in the open string ﬁeld theory is intriguing. However, it remains to be seen whether string ﬁeld theory is a useful tool beyond perturbation theory. It has not played a role in the recent nonperturbative progress based on spacetime supersymmetry, which will be described in chapter 14. It is quite possible that strings, or string ﬁelds, are simply not the correct degrees of freedom for a nonperturbative formulation of ‘string’ theory. 9.7

Large order behavior

There is an interesting result about the large order behavior of string perturbation theory. Let us start with ﬁeld theory. Consider the integral

∞

y2 y3 dy exp − − λ = λ−1 2! 3! −∞

∞

1 z2 z3 dz exp − 2 + λ 2! 3! −∞

,

(9.7.1)

where z = yλ. This integral diverges for λ = 0, but for now we can ignore that and calculate it as a formal expansion in powers of λ, ∞ (−λ)n ∞ n=0

6n n!

−∞

dy y 3n exp(−y 2 /2) = (2π)1/2

∞

λ2k C2k ,

(9.7.2)

k=0

with C2k =

2k Γ(3k + 1/2) . π 1/2 32k (2k)!

(9.7.3)

This integral is the zero-dimensional limit of a scalar ﬁeld theory, with a mass term and cubic coupling. The expansion in λ can be represented in terms of Feynman graphs, with propagator 1 and cubic vertex −λ. The constant C2k then simply counts the total number of vacuum graphs with 2k cubic vertices, or equivalently k + 1 loops, weighted by the appropriate symmetry factor. For example, the reader can check that there are two vacuum graphs with two vertices, with symmetry groups of order 8 and 5 1 = 18 + 12 . To get the large order behavior, we use 12, and indeed C2 = 24 n+1/2 e−n . Then Stirling’s approximation n! ≈ n C2k ≈ k k ≈ k! ,

(9.7.4)

where the approximation holds up to factors of the form k a and bk . One can also include a quartic term λ2 y 4 /4!, which makes the integral converge. The power of λ is determined by requiring that all terms in the action for z be proportional to λ−2 ; equivalently, all k + 1 loop vacuum graphs are weighted by λ2k . This additional term does not substantially change the result; the estimate (9.7.4) still holds to the given accuracy.

316

9 Higher order amplitudes

When considering ﬁeld theory, the propagator and vertices become more complicated. However, the dominant behavior at large orders usually comes from simple counting of graphs — the propagators and vertices only contribute factors of order bk , unless there is unusual IR or UV behavior. The series is of the form ∞

λ2k k!f2k ,

(9.7.5)

k=0

where f2k ≈ k a bk varies more slowly than k!. This diverges unless f2k has unexpected cancellations. For small λ the terms become smaller for a while and then grow as the k! dominates. The smallest terms in the sum are of order exp[−O(1/λ2 )] .

(9.7.6)

This is then an indication of the accuracy of perturbation theory as an asymptotic series. Indeed, eﬀects like instantons and dynamical symmetry breaking, which vanish to all orders of perturbation theory, are typically of this order. For example, the large order behavior of the series (9.7.2) is related to the ‘instanton’ (stationary point) at z = −2, with action 2/3λ2 . In string theory, the Feynman graph decomposition of moduli space makes it possible to establish the same bounds. It has been shown explicitly that the series is of the form (9.7.5) with f2k positive and bounded below by bk . Thus the series is divergent, and is not even Borel summable. Borel summation rewrites the series (9.7.5) as the Laplace transform of a more convergent series ∞ 0

dt exp(−t)

∞

(tλ2 )k f2k .

(9.7.7)

k=0

However, the positivity of the f2k implies that this new sum, even if it converges, has poles on the positive t-axis, so that the integral is ambiguous. Thus, we should expect that string theories will have interesting nonperturbative phenomena. This is not surprising because even in the low energy limit they reduce to nontrivial ﬁeld theories. The estimate (9.7.4) for closed string theory is a lower bound, which can be obtained even keeping only the tree-level cubic vertex V0,3 . Because the closed string vertex receives corrections from Riemann surfaces of all genus, one might wonder whether perturbation theory grows even more rapidly. One way to see that in fact it does is to recall that the open string ﬁeld theory contains all open plus closed string Riemann surfaces with at least one boundary. Let us consider the surfaces with one boundary, in the region where the boundary has shrunk to a point. This limit of open string moduli space is equivalent to the moduli space for closed surfaces

9.8 High energy and high temperature

317

with one vertex operator. One can show that this region is not vastly smaller than the full open string moduli space, so the same estimate holds, the perturbation series going as ∞

go2k O(k!) =

k=0

∞

gck O(k!) .

(9.7.8)

k=0

The genus-g term, proportional to gc2g , has a coeﬃcient of order (2g)!. This is larger than the g! characteristic of ﬁeld theory, which was also the lower bound obtained from V0,3 alone. Evidently the largest part of the large order amplitudes is contained within the vertices Vg,n and comes from the interior of moduli space. One then expects nonperturbative eﬀects of order exp[−O(1/gc )] ,

(9.7.9)

much larger at weak coupling than the exp[−O(1/gc2 )] characteristic of ﬁeld theory. This large order behavior was ﬁrst observed in some simpliﬁed models to be discussed in section 9.9. As we will see in chapter 13, these eﬀects are now known to be related to D-branes, at least in some supersymmetric string theories. 9.8

High energy and high temperature

In order to obtain a better understanding of string dynamics, one might look at the behavior in diﬀerent limits. In this section we consider various high energy limits of string theory. Hard scattering Consider again the hard scattering behavior, which we obtained from the Veneziano and Virasoro–Shapiro amplitudes by Stirling’s approximation. Go back to the integral form (6.6.5), 8πigc2 (2π)26δ 26 ( α

i ki )

C

d2 z4 |z4 |−α u/2−4 |1 − z4 |−α t/2−4 .

(9.8.1)

In the hard scattering region s → ∞ with t/s ﬁxed, the exponents are large and the integral is dominated by the stationary points of the exponent.4 The saddle point, for s, t, u 1, is given by u t ∂ u ln |z4 |2 + t ln |1 − z4 |2 = + = 0, ∂z4 z4 z4 − 1

4

(9.8.2)

As usual, the integral diverges at z4 → ∞ and is deﬁned by analytic continuation. The saddle point approximation can be justiﬁed for the continuation.

318

9 Higher order amplitudes

with solution z4 = −u/s; recall that s + t + u = O(1). The saddle point approximation then gives an amplitude S ≈ exp[−α (s ln s + t ln t + u ln u)/2] ,

(9.8.3)

as given in eq. (6.6.13). Let us also back up one step, from the integral over z4 to the path integral over X µ . This is dominated by the saddle point Xclµ (σ) = i

µ ki G (σ, σi ) ,

(9.8.4)

i

where the σi are the vertex operator positions. Recall that the Green’s function is proportional to α . At high energy the size of the scattering region grows linearly with energy, δX ≈ α k. In ﬁeld theory, higher energy hard scattering probes shorter distances, δX ≈ 1/k by the uncertainty principle. In string theory this is not the case beyond the string scale. It appears to be impossible in string perturbation theory to probe distances below this scale, giving rise to the idea that it is a limiting distance in spacetime. The eﬀective lower limit on compactiﬁcation R ≈ α1/2 also supports this. However, we will see in chapter 13 that recent advances in nonperturbative string theory reveal the existence of shorter length scales. At general orders in the string loop expansion, the dominant term in the amplitude will be

exp −

ki · kj G (σi , σj ) .

(9.8.5)

i<j

We need to ﬁnd the saddle points of the exponent, with respect to both the vertex operator positions and the moduli of the Riemann surface. This is like an electrostatics problem: the exponent is the energy of n point charges kiµ on a Riemann surface, where µ labels d diﬀerent electrostatic ﬁelds and µ = 0 has the opposite sign in the energy. The background charge density vanishes by momentum conservation, and the renormalized self-energy is negligible because ki2 remains on-shell and small. Remarkably, one can ﬁnd what is probably the dominant saddle point at each genus. Consider the Riemann surface (z1 − a1 )(z1 − a2 ) . (9.8.6) z2N = (z1 − a3 )(z1 − a4 ) Like the hypergeometric curve (9.2.9), it is a subspace of S2 × S2 . It can be regarded as N sheets glued together at branch cuts, and has genus N − 1. For the four-point amplitude, if the charges are placed at the branch cuts ai then the amplitude is stationary with respect to their positions — the surface is ‘vertical’ at these points, ∂z2 /∂z1 = ∞, and the symmetry z2 → exp(2πi/N)z2 requires the vertical component of the gradient to vanish. It remains to extremize with respect to the positions ai , but this

9.8 High energy and high temperature

319

is easy: the electric ﬁeld is the same as on the sphere with charges at z = ai , except that it is divided between N sheets. Thus the energy is N · N −2 = N −1 times that on the sphere. Three of the positions can be ﬁxed by M¨ obius invariance, for example to 0, 1, ∞, and the fourth is given by the same condition (9.8.2) as on the sphere. Thus at genus g, Sg ≈ exp[−α (s ln s + t ln t + u ln u)/2(g + 1)] .

(9.8.7)

With additional eﬀort one can include the Gaussian ﬂuctuations around the saddle, which determine the coeﬃcient of Sg . For the N-sheeted saddle points, Xclµ is simply divided by N = g + 1. There is a check that this result makes sense. In the small-angle region −t/s 1 it becomes exp[α t ln s/2(g + 1)] .

(9.8.8)

Now, t1/2 is the total momentum transferred from the incoming particle 1 to the ﬁnal particle 3. At small angles it is plausible that the genus-g scattering process can be decomposed into g + 1 successive scatterings. Each separate scattering has the same center of mass s, while the ti are constrained so that the total momentum transfer is t1/2 . The individual scatterings go as exp(α ti ln s/2). It is then easy to check that it is most 1/2 1/2 eﬃcient to take the momentum transfers to be equal, ti = t /(g + 1), leading to the result (9.8.8) for the product of the amplitudes. This is diﬀerent from particle scattering, where it is most eﬃcient to transfer momentum in a small number of hard scatterings. For excited string states, the exponential part of the vertex operator is the same as for the tachyon. The same saddle point then dominates, with Xclµ inserted in the vertex operators. Since this same saddle point dominates for all states and, up to the factor g + 1, all orders, there are linear relations between the amplitudes for diﬀerent states that hold for all genus and therefore plausibly for the exact amplitudes. In quantum ﬁeld theory, high energy scattering often reveals symmetries that are not obvious at low energy. It has been suggested that the relations that emerge here are of the same sort. It is not clear to what extent it makes sense to sum the series (9.8.7). It is interesting to contrast the result here with that in the previous section. In both cases it is the interior of moduli space that is producing the interesting physics (in contrast to the ﬁeld-theoretic behavior found near the boundary). However, the large order behavior came from the whole of moduli space, dominated in particular by its large volume. The high energy behavior at ﬁxed genus is dominated by a single point.

320

9 Higher order amplitudes Regge scattering

The analysis of the Regge limit can also be carried to higher orders. We will not do this, but will develop an interesting interpretation of the tree-level Regge physics. Regge scattering is less familiar to most physicists than hard scattering. It is a gentle probe of a highly boosted system. When a system is boosted, time dilation slows down its internal processes. One might then expect to see more of the virtual degrees of freedom as one goes to higher energies. Let us in particular consider the quantum ﬂuctuations of a single string, in particular the root mean size of the ground state. The mode expansion gives

0|[X 1 (σ) − x1 ]2 |0 =

∞ α

2n2

0|(αn α−n + ˜αn ˜α−n )|0

n=1 ∞

=α

1 . n n=1

(9.8.9)

This is divergent but has no direct physical signiﬁcance. A measurement on a time-scale δτ is sensitive only up to modes of frequency less than δτ−1 , essentially n α−1/2 < δτ−1 . The logarithmic divergence becomes α ln(α1/2 /δτ), and the size is the square root of this. This growth in the eﬀective size is directly visible in the Regge amplitude (6.6.12) at small t, Γ(−α t/4 − 1) s2+α t/2 Γ(α t/4 + 2)

s2 q 2 α ∼ exp − ln(α s) , t 2

(9.8.10)

where q 2 = −t is the momentum transfer. This is the gravitational amplitude s2 /t modiﬁed by a form-factor whose Fourier transform corresponds to an object of size (α ln α s)1/2 . From the point of view of one string the other has a boost of order α s, so the time resolution of its internal processes is inverse to this, in agreement with the earlier estimate for the size. The square root of a logarithm is a very slow function and ordinarily would be of little importance, but it has been argued that this behavior is very signiﬁcant for strings near black holes. In chapter 14 we will revisit, from a diﬀerent point of view, the idea that strings reveal their true nature when very highly boosted. High temperature A similar limit with interesting stringy behavior is high temperature. From the generic behavior (7.2.30) of the partition function at high weight h,

321

9.8 High energy and high temperature

and the relation m2 = 4(h − 1)/α , the density of string states n(m) satisﬁes ∞ 0

dm n(m) exp(−α πm2 ) ≈ exp(4π/) ,

(9.8.11)

implying n(m) ≈ exp(4πmα1/2 ). The thermal partition function of a single string is then ∞

dm exp(4πmα1/2 ) exp(−m/T ) .

(9.8.12)

0

Because the density of states grows exponentially, the partition function (9.8.12) diverges for temperatures greater than the Hagedorn temperature TH =

1 . 4πα1/2

(9.8.13)

One can understand the Hagedorn divergence as follows. The energy of a long string is proportional to its length. So also is its entropy, being extensive. At low temperature the energy dominates, but at high temperature the entropy dominates and it is favorable to produce strings of arbitrary length. In an ensemble of such strings the density will become large, and the interactions will be important near the transition. This divergence suggests a phase transition. Indeed, in the strong interaction there is a similar growth in the density of states of highly excited hadrons, and there is a phase transition from a low temperature hadron phase to a high temperature quark–gluon phase. It is interesting to wonder whether the Hagedorn divergence indicates some analogy to the quark–gluon structure in the fundamental string, but there is no clear answer. To conclude, let us discuss the partition function of free strings in somewhat more detail. The free energy density of a real scalar ﬁeld of mass m in dimension d is 2

dd−1 k ln[1 − exp(−ωk /T )] (2π)d−1 ∞ ∞ dt m2 t r2 −d/2 =− exp − . (2πt) − t 2 2T 2 t 0 r=1

F(T , m ) = T

(9.8.14)

In the second form, one can readily sum over the string spectrum. Following the derivation of the vacuum amplitudes (7.3.12) and (7.3.6), the free energy density in 26 ﬂat dimensions is F(T ) = −

R

∞ dτd¯τ r2 (4π 2 α τ2 )−13 |η(τ)|−48 exp − 2τ2 4πT 2 α τ2 r=1

.

(9.8.15)

322

9 Higher order amplitudes

The integral runs over the full region R:

τ1 ≤

1 , 2

|τ2 | > 0 .

(9.8.16)

The derivation from a modular-invariant path integral is given as exercise 9.7. There is a possible divergence as τ2 → 0; the integrand is largest when the real part of τ is zero, so let τ = iτ2 . From the modular transformation (7.2.44), the asymptotic behavior of the eta function in this limit is −1/2

η(iτ2 ) = η(i/τ2 )τ2 ≈ exp(−π/12τ2 ) as τ2 → 0 .

(9.8.17)

The τ integral thus converges for T < TH and diverges for T > TH . The following curious observation has been made. In ﬁeld theory the partition function is given by a Euclidean path integral, with time having period 1/T . The same is true in string theory (exercise 9.7). T -duality should then relate the high and low temperature behaviors, and in fact the Hagedorn temperature is half the self-dual value. Deﬁne ˜ ) = F(T ) + ρ0 , F(T

(9.8.18)

the thermal part (9.8.15) of the free energy plus the vacuum part (7.3.6). ˜ )/T , so T -duality implies The logarithm of the path integral is −F(T 1˜ T ˜ F(4TH2 /T ) . F(T ) = T 4TH2

(9.8.19)

This relates the partition function below the Hagedorn temperature to that above it. The Hagedorn divergence for T > TH is a reﬂection of the tachyon divergence in the low temperature theory, but a similar formal result holds for the supersymmetric string, which has no tachyonic divergence. If we take this relation seriously, we obtain 2 ˜ ) → T ρ0 F(T 4TH2

(9.8.20)

as T → ∞. If this is a measure of the number of high energy degrees of freedom, they are very few. A single quantum ﬁeld in d spacetime dimensions has a free energy of order T d , so it is as though the high energy limit of string theory looks like a ﬁeld theory in two spacetime dimensions. This is entirely unlike the exponentially growing spectrum below the Hagedorn transition, and may be a clue to the nature of the phase transition.

9.9 Low dimensions and noncritical strings 9.9

323

Low dimensions and noncritical strings

Field theories often simplify as the number of spacetime dimensions is reduced, and these low-dimensional theories have been useful models for understanding dynamics. We might try to do the same in string theory, but string theory requires a critical dimension. We could compactify, but this does not reduce the number of degrees of freedom: the string still oscillates in the compact directions. We have, however, encountered a consistent background in which the string actually does move in fewer dimensions. This is the linear dilaton background (3.7.18), Gµν (x) = ηµν ,

Φ(x) = Vµ xµ ,

Bµν (x) = 0 ,

1/2

(9.9.1a)

26 − D . (9.9.1b) 6α For D < 26 the dilaton gradient is spacelike and we have taken it to lie in the 1-direction. The eﬀective string coupling eΦ is a function of x1 , diverging at large x1 . This makes it hard to use string perturbation theory, but fortunately there is a slightly diﬀerent background for which perturbation theory is good. Inserting the background (9.9.1) into the tachyon action (6.6.16) leads to the linearized tachyon ﬁeld equation, Vµ = δµ1

4 T (x) = 0 . (9.9.2) α This is the condition for the exponential vertex operator to be Weylinvariant, with the linear dilaton energy-momentum tensor (2.5.1). The solutions are 2−D T (x) = exp(q · x) , (q − V )2 = . (9.9.3) 6α − ∂µ ∂µ T (x) + 2V µ ∂µ T (x) −

If we look for solutions which, like the dilaton, depend only on x1 , then

q1 = α ± =

26 − D 6α

1/2

±

2−D 6α

1/2

.

(9.9.4)

For D > 2 this is complex, so the tachyon oscillates, but for D ≤ 2 one gets a real exponential (times a linear function at the critical value D = 2). Of course, the tachyon ﬁeld equation will have nonlinear corrections. It can be shown that this does not aﬀect the qualitative form of the background, except that for rather subtle reasons only the solution α− leads to a nonsingular background. For D ≤ 2 the term ‘tachyon’ for the string ground state is actually a misnomer, because this ﬁeld is stable (exercise 9.8). Also for D ≤ 2, the real exponential T (x) = T0 exp(α− x1 ) prevents propagation to the region

324

tachyon background

weak coupling

9 Higher order amplitudes

Fig. 9.12. Spacetime with dilaton and tachyon backgrounds. The interaction strength grows toward the right but is cut oﬀ by an eﬀective potential from the tachyon background.

of large positive x1 where the coupling diverges. To see this, consider the world-sheet action

1 2 1/2 ab µ ν 1 1 Sσ = d σ g g η ∂ X ∂ X + α RV X + T exp(α X ) . µν a 1 0 − b 4πα M (9.9.5) At large x1 the exponential tachyon dominates the linear dilaton and the path integral is suppressed, corresponding to an eﬀective potential for the string due to the tachyon background. Physically one has an asymptotic region x1 → −∞, where the coupling goes to zero. Strings propagating toward positive x1 interact with one another and reﬂect oﬀ the potential and back to the asymptotic region. The maximum eﬀective coupling is set by the energy and the strength T0 of the potential. This picture, shown in ﬁgure 9.12, is a bit exotic, but in the interest of ﬁnding a solvable model it is worth pursuing. A model with rather similar physics, the static model, played a role in unraveling the nature of quantum ﬁeld theory. For D > 2 the physical eﬀect of the tachyon background is less clear, though it has been argued that it still acts as a barrier. However, the focus has been on D ≤ 2 for another reason, namely that these theories are actually solvable. Let us ﬁrst observe that when we quantize the string, the gauge symmetries take away two families of oscillators, so that in D ≤ 2 none will be left — there is only the center-of-mass motion, a single

9.9 Low dimensions and noncritical strings

325

scalar.5 Nevertheless, to solve the string theory directly is still quite hard. First one has to evaluate the world-sheet path integral with action (9.9.5). This is no longer a free theory, because of the exponential tachyon ﬁeld. It is known as the Liouville ﬁeld theory. It is classically integrable, and with substantial eﬀort some quantum amplitudes have been determined. However, to solve the string theory one would still need to integrate the result at each genus over the moduli space of Riemann surfaces; we have seen that this is a complicated space. Remarkably, the full amplitude has been determined, to all orders, by very diﬀerent means. To describe this, we must ﬁrst introduce another idea. Noncritical strings Consider a statistical mechanical system deﬁned by a sum over twodimensional surfaces in a d-dimensional Euclidean space. Let us also suppose that the surfaces have some internal degree of freedom that can be interpreted as a measure of length. The natural ﬁelds to describe the system are the embedding X µ (σ) and a metric gab (σ). However, there is no particular reason for the length to drop out of the problem — that is, for the theory to be Weyl-invariant. This is therefore called the noncritical string. The lowest-dimensional action (or Boltzmann weight, since we are thinking of the path integral in the context of statistical mechanics) for this system would then be the Polyakov action with an additional worldsheet cosmological constant term, which in the critical string is forbidden by Weyl invariance, 1 S= 4πα

M

d2 σ g 1/2 g ab ∂a X µ ∂b Xµ + µ .

(9.9.6)

Without Weyl invariance, we can use diﬀ invariance to ﬁx the metric partially, but one degree of freedom remains: gab (σ) = exp(2φ(σ)) gˆ ab (σ) ,

(9.9.7)

where gˆ ab is a gauge choice and the scale φ of the metric is still pathintegrated. The Faddeev–Popov determinant for this gauge-ﬁxing is the same as in eq. (3.4.19), the Weyl-ﬁxing contribution to the latter being trivial. Inserting the metric (9.9.7), for which the curvature is

ˆ 2φ , g 1/2 R = gˆ 1/2 Rˆ − 2∇

(9.9.8)

and dropping a ﬁeld-independent term, the action plus Faddeev–Popov 5

At special momenta one ﬁnds additional physical states. These can be interpreted as unbroken symmetries of string ﬁeld theory, and likely play a role in the solvability of the theory.

326

9 Higher order amplitudes

determinant becomes

13α ab ˆ gˆ ∂a φ∂b φ + Rφ . 3 M (9.9.9) Note that the Faddeev–Popov determinant has provided a kinetic term for φ. Comparing this with the dilaton–tachyon action (9.9.5) we see a marked resemblance. If we let d = D − 1 but interpret the new ﬁeld φ as x1 , and if we rename gab → gˆ ab in the action (9.9.5) (since it can be fully gauge-ﬁxed), then all the same terms appear. Some coeﬃcients are a bit diﬀerent — there is no rescaling of φ that makes them all agree — but in fact we claim that these are the same ﬁeld theory. The point is that the critical string path integral (9.9.5) is deﬁned using gˆ ab as the metric, and the noncritical string (9.9.9) using exp(2φ)gˆ ab . A careful treatment of the Weyl transformation of the measure then converts one to the other. One can understand this quite easily by noting that only the full metric (9.9.7) appears in the deﬁnition of the noncritical string, so nothing can change under

S=

1 4πα

d2 σ gˆ 1/2 gˆ ab ∂a X µ ∂b Xµ + µ exp 2φ +

gˆ ab (σ) → exp(2ω(σ)) gˆ ab (σ) , φ(σ) → φ(σ) − ω(σ) .

(9.9.10a) (9.9.10b)

The noncritical string thus acquires a Weyl invariance, and this is enough to ﬁx the various coeﬃcients, other than the free parameter µ. It may seem odd that Weyl-invariant and noninvariant theories can be equivalent. The point is that in ﬁeld theory we can always add extra ‘fake’ gauge symmetries if we also add extra ﬁelds that we just use to ﬁx the gauge. A gauge symmetry after all is just a redundancy of description, and we can always adopt a more redundant description. For the symmetry (9.9.10) it is clear that we can just cancel the Weyl symmetry against the ﬁeld φ, but for general string backgrounds (for example ﬂat space, where the embedding coordinates are Weyl-invariant) this is not possible and the Weyl invariance is ‘real.’ In any case we are now nearing the punch line. One can approximate the noncritical string by a sum of triangulated surfaces, the number of edges between two points being a measure of length. These triangulated surfaces resemble large Feynman diagrams, and in fact one can ﬁnd a quantum system, the matrix model, for which this is the diagram expansion. The ﬁelds of the matrix model are N × N matrices. Taking N → ∞ and adjusting other couplings, there is (at least for D ≤ 2) a critical point dominated by large smooth surfaces, so we recover the noncritical string. Moreover, the matrix model is solvable (again for D ≤ 2) by diagonalizing the matrices. In the end one has a rather simple system (noninteracting

Exercises

327

fermions moving in the space of eigenvalues) which by retracing the above steps allows the calculation of string amplitudes to all genus. Because these methods are so far aﬁeld from the rest of the book, we will have to leave the details to the references. In any case they have not yet been useful at D > 2 or in supersymmetric theories. (However, we will see in chapter 14 that a somewhat diﬀerent theory of large-N matrices has recently been proposed as a nonperturbative string theory.) We should mention, though, at least one important lesson from these models. This is the (2g)! behavior of perturbation theory, which was seen ﬁrst in the explicit matrix model solution and then argued to hold in all string theories. In the matrix model the corresponding exp[O(−1/gc )] eﬀect involves tunneling of the fermions in the space of eigenvalues. It is not clear whether this can be related to the exp[O(−1/gc )] eﬀects in higher-dimensional string theories. Exercises 9.1 (a) Use a conformal transformation to show that the OPE (9.1.2) and expectation value (9.1.3) imply the inner product (9.1.29). ˆ j (k) , verify the second (b) By taking matrix elements in the basis |V equality in equation (9.1.28). 9.2 (a) Write the four-point amplitude on the torus in operator form, parallel to eq. (9.1.17) for the sphere, but now involving a trace. ˆ 1 = {QB , X}, then (b) Show that if one of the vertex operators is null, V the trace is equal to total derivatives with respect to the vertex operator positions plus total derivatives with respect to τ and ¯τ. 9.3 (a) For the torus with four vertex operators, use the result of part (a) of the previous problem to study the unitarity in the limit that w1 , w2 , and w3 − w4 are ﬁxed and w3 and 2πτ − w3 approach i∞. This is two handles degenerating, and unitarity relates the amplitude to the product of two tree-level four-point amplitudes. (b) Repeat in particular the demonstration that the sum over the full ˆ and that over the physical Hilbert space are equal. Show Hilbert space H that this requires cancellation between intermediate states of diﬀerent ghost numbers. This cancellation is a stringy generalization of the cancellation between Faddeev–Popov ghosts and longitudinal gauge bosons in gauge theory. 9.4 Verify the assertion in eq. (9.3.7) about the region covered by the Feynman graph of ﬁgure 9.6(a), and ﬁnd the other two regions covered by this graph. 9.5 (a) For the terms (9.6.7) in the expansion of the string ﬁeld, write out explicitly the gauge transformation (9.6.2) and the action (9.6.3).

328

9 Higher order amplitudes

(b) Eliminate auxiliary ﬁelds and ﬁelds that can be completely gauged away, and show that the resulting gauge transformation and action for Aµ have the expected form. 9.6 For f2k = k a bk , show that the smallest term in the series (9.7.5) is of the order of exp(−C/g 2 ). Work to suﬃcient accuracy to ﬁnd C. For the series (9.7.2), compare with the instanton action. 9.7 In quantum theory, the Euclidean path integral with Euclidean time ˜ )/T ]. having period T −1 gives the thermal partition function exp[−F(T Show that for the string path integral this agrees with the result (9.8.15). The one subtlety is an interplay between the region of integration and the sum over windings (Reference: McClain & Roth, (1987).) 9.8 For the tachyon wave equation (9.9.2) in the linear dilaton background (9.9.1), show that any ﬁnite energy initial conﬁguration has stable (oscillatory) evolution provided that D ≤ 2. 9.9 Study the physical state conditions (either in OCQ or BRST form) for the graviton level of the closed string in the D = 2 linear dilaton background. Let the momenta take complex values. You should ﬁnd that the physical spectrum is trivial at generic momenta but that at two momenta there are nontrivial physical states. 9.10 Derive eq. 9.8.14. Hint: in the second line introduce an integral of k such that m2 becomes m2 + k2 . Then integrate over t (the result is in terms of elementary functions. Finally, the sum on r is the series expansion of the log in the ﬁrst line.

Appendix A A short course on path integrals

Path integrals are a powerful means of representing quantum theories, especially on surfaces of nontrivial topology. The introduction we present here is similar to that which the reader will ﬁnd in any modern ﬁeld theory text. We include it in order to emphasize certain ideas that we will need, such as the relation between the path integral and Hilbert space formalisms and the use of operator equations inside the path integral. A.1

Bosonic ﬁelds

Consider ﬁrst a quantum mechanics problem, one degree of freedom with ˆ p, q), ˆ where Hamiltonian H(ˆ ˆ pˆ ] = i¯h . [q,

(A.1.1)

Throughout the appendix operators are indicated by hats. A basic quantity of interest is the amplitude to evolve from one q eigenstate to another in a time T , ˆ /¯h)|qi .

qf | exp(−iHT

(A.1.2)

In ﬁeld theory it is generally convenient to use the Heisenberg representation, where operators have the time dependence ˆ = exp(iHt/¯ ˆ h) A ˆ exp(−iHt/¯ ˆ h) . A(t)

(A.1.3)

ˆ The state |q, t is an eigenstate of q(t), ˆ q(t)|q, t = |q, t q .

(A.1.4)

In terms of the t = 0 Schr¨ odinger eigenstates this is ˆ h)|q . |q, t = exp(iHt/¯ In this notation the transition amplitude (A.1.2) is qf , T |qi , 0 . 329

(A.1.5)

330

Appendix A

qf

...

T

t2 t1 0

qi

Fig. A.1. Transition amplitude broken down into time steps. The dashed line suggests a piecewise linear path, which is one of many ways to deﬁne path integration.

By inserting a complete set of states, we can write the transition amplitude as a coherent sum over all states q through which the system might pass at some intermediate time t:

qf , T |qi , 0 =

dq qf , T |q, t

q, t|qi , 0 .

(A.1.6)

Divide the time interval further into N steps as in ﬁgure A.1, tm = m ,

= T /N ,

(A.1.7)

and corresponding to each intermediate time insert a complete set of states

qf , T |qi , 0 =

dqN−1 . . . dq1

N−1

qm+1 , tm+1 |qm , tm .

(A.1.8)

m=0

Translate back to Schr¨ odinger formalism and introduce an integral over intermediate momenta, ˆ h)|qm

qm+1 , tm+1 |qm , tm = qm+1 | exp(−iH/¯

=

ˆ h)|qm . dpm qm+1 |pm

pm | exp(−iH/¯

(A.1.9)

ˆ with all pˆ to the left and all qˆ to By commuting, we can always write H the right, so that ˆ p, q)|q ˆ m = H(pm , qm ) pm |qm .

pm |H(ˆ

(A.1.10)

Use this to evaluate the matrix element in eq. (A.1.9). From expanding the exponential there will be terms of order 2 and higher that have qˆ to the

331

A.1 Bosonic ﬁelds left of pˆ ; drop these. Then to this order qm+1 , tm+1 |qm , tm becomes

dpm exp[−iH(pm , qm )/¯ h] qm+1 |pm

pm |qm

=

!

i

dpm exp − H(pm , qm ) − pm (qm+1 − qm ) + O(2 ) 2π¯ h ¯ h

Thus

qf , T |qi , 0 ≈

→

.

(A.1.11)

dpN−1 dp0 dp1 dqN−1 . . . dq1 2π¯h 2π¯h 2π¯h

"

i N−1 × exp − H(pm , qm ) − pm (qm+1 − qm ) ¯h m=0 ! T i

"

dt [ p˙ q − H(p, q) ] . (A.1.12) h¯ 0 In the last line we have taken a formal → 0 limit, so the integral runs over all paths p(t), q(t) with given q(T ) and q(0). This is the Hamiltonian path integral (or functional integral), the sum over phase space paths weighted by exp(iS /¯h) with S the action in Hamiltonian form. Let us integrate over p(t), making ﬁrst the approximation that the integral is dominated by the point of stationary phase, 0=

[dpdq] exp

∂ ∂ [ p˙ q − H(p, q) ] = q˙ − H(p, q) . ∂p ∂p

(A.1.13)

Solving for p in terms of q and q˙ and recalling that L = p˙ q − H, the stationary phase approximation gives

qf , T |qi , 0 =

T i

[dq] exp

¯h

dt L(q, q˙) .

(A.1.14)

0

ˆ = 1 pˆ 2 + V (q), ˆ so that the Often the momentum integral is Gaussian, H 2 stationary phase approximation is exact up to a normalization that can be absorbed in the measure [dq]. In fact, we can do this generally: because the relation (A.1.13) between p and q˙ is local, the corrections to the stationary phase approximation are local and so can be absorbed into the deﬁnition of [dq] without spoiling the essential property that [dq] is a product over independent integrals at each intermediate time. The result (A.1.14) is the Lagrangian path integral, the integral over all paths in conﬁguration space weighted by exp(iS /¯h), where now S is the Lagrangian action. We have kept ¯h so as to discuss the classical limit. As ¯h → 0, the integral is dominated by the paths of stationary phase, δS /δq(t) = 0, which is indeed the classical equation of motion. Henceforth, h¯ = 1. The derivation above clearly generalizes to a multicomponent q i . One can always write a quantum ﬁeld theory in terms of an inﬁnite number of qs, either by making the spatial coordinate x discrete and taking a limit,

332

Appendix A

or by expanding the x-dependence in terms of Fourier modes. Thus the above derivation of the path integral representation applies to bosonic ﬁeld theory as well. The derivation of the path integral formula treated the continuum limit → 0 rather heuristically. Renormalization theory is the study of this limiting process, and is rather well understood. The limit exists provided the Lagrangian satisﬁes certain conditions (renormalizability) and provided the coeﬃcients in the Lagrangian (bare couplings) are taken to vary appropriately as the limit is taken. The result is independent of various choices that must be made, including the operator ordering in the Hamiltonian, the precise deﬁnition of the measure, and the diﬀerences between regulators (the limit of piecewise linear paths suggested by the ﬁgure versus a frequency cutoﬀ on the Fourier modes, for example).1 Being local eﬀects these can all be absorbed into the bare couplings. Actually, we will evaluate explicitly only a few Gaussian path integrals. We use the path integral primarily to derive some general results, such as operator equations and sewing relations. Relation to the Hilbert space formalism Starting from the path integral one may cut it open to recover the Hilbert space of the theory. In equation (A.1.6), if one writes each of the amplitudes as a path integral one ﬁnds

T

qf ,T

[dq]q ,0 exp i

=

i

dq

dtL 0 qf ,T

T

[dq]q,t exp i

dtL t

q,t

t

[dq]q ,0 exp i i

dtL

. (A.1.15)

0

That is, the path integral on [0, T ] breaks up into separate path integrals on [0, t] and [t, T ], plus an ordinary integral over q(t). The boundary conditions on paths are denoted by super/subscripts. Note the restriction that T ≥ t ≥ 0. One can also replace the intermediate integral with a sum over any complete set of states. This cutting (or, inversely, sewing) principle is particularly useful on the surfaces of nontrivial topology that arise in string theory. There one may cut the surface open along many diﬀerent closed curves; diﬀerent choices give diﬀerent Hilbert space representations of a given amplitude. This is known as world-sheet duality. Consider now a path integral with the additional insertion of a factor 1

As another example, one can freely rescale ﬁelds. The Jacobian is formally a product of separate Jacobians at each point; its logarithm is therefore a sum of a local expression at each point. Regulated, this is just a shift of the bare couplings.

333

A.1 Bosonic ﬁelds of q(t), where 0 < t < T . Then, using the cutting relation (A.1.15),

qf ,T

[dq]q ,0 exp(iS )q(t) = i

=

dq qf , T |q, t q q, t|qi , 0

ˆ dq qf , T |q(t)|q, t q, t|qi , 0

ˆ = qf , T |q(t)|q i , 0 .

(A.1.16)

ˆ Thus, q(t) in the functional integral translates into q(t) in the matrix element. For a product of two insertions q(t)q(t ) (which is equal to q(t )q(t), since these are simply variables of integration), where both t and t are in the range [0, T ], one ﬁnds by the same method that

qf ,T

ˆ q(t ˆ )]|qi , 0 . [dq]q ,0 exp(iS )q(t)q(t ) = qf , T |T[q(t)

(A.1.17)

Here, T denotes the time-ordered product ˆ B(t ˆ ) + θ(t − t)B(t ˆ )A(t). ˆ ˆ B(t ˆ )] = θ(t − t )A(t) T[A(t)

(A.1.18)

i

Because of the way the path integral is built out of successive inﬁnitesimal time slices, two or more insertions in the path integral will always correspond to the time-ordered product of operators in the matrix element. The equation of motion is δS /δq(t) = 0. Let us see how this works inside the path integral. Consider this equation in a path integral with other insertions F,

qf ,T

[dq]q ,0 i

δ [dq]q ,0 exp(iS ) F i δq(t) qf ,T δF . (A.1.19) = i [dq]q ,0 exp(iS ) i δq(t)

δS exp(iS ) F = −i δq(t)

qf ,T

We have integrated by parts in the second line.2 If none of the ﬁelds in F are at time t, the right-hand side vanishes and the equation of motion holds in the path integral. We therefore say that the equation of motion holds as an operator equation. This is a technique that we will use extensively: an operator equation is one which holds when inserted into any path integral that has no other ﬁelds coincident with those involved in the equation. Translating to Hilbert space language, we can use the other insertions F to set up arbitrary initial and ﬁnal states, so this corresponds 2

The functional derivative is deﬁned as δ q(t ) = δ(t − t ) . δq(t) It can be regarded as the limit of the ordinary derivative 1 ∂ , ∂qm

m = t/ .

In particular, one can integrate a functional derivative by parts, as for an ordinary derivative.

334

Appendix A

to the ordinary notion that an operator equation is one that should hold for general matrix elements. When additional insertions are present at t, operator equations inside the path integral hold only up to additional delta-function source terms. For example, let F = q(t ) . . . , where t is near t and the other ﬁelds in ‘. . .’ are far away. Then eq. (A.1.19) becomes -

q(t )

δS ... δq(t)

.

= i δ(t − t ) . . .

.

(A.1.20)

This is an example of the Schwinger–Dyson equation, the equation of motion for the expectation value of a product of ﬁelds. The same result written as an operator equation is q(t )

δS = iδ(t − t ) . δq(t)

(A.1.21)

As an explicit example, take the harmonic oscillator, 1 S = 2

dt (˙ q2 − ω2q2) ,

δS = −¨ q (t) − ω 2 q(t) . δq(t) Then the operator equations we have derived are 2 ∂

(A.1.22a) (A.1.22b)

+ ω 2 q(t) = 0

(A.1.23)

+ ω 2 q(t)q(t ) = −iδ(t − t ) .

(A.1.24)

∂t2 and 2 ∂

∂t2

In Hilbert space language these become 2 ∂ 2 ∂

+ω

∂t2

2

ˆ =0, q(t)

ˆ q(t ˆ )] = −iδ(t − t ) . + ω 2 T[q(t)

(A.1.25a) ∂t2 The ﬁrst is the Heisenberg equation for the ﬁeld operator, while the delta function in the second comes from the time derivative acting on the step functions in the time-ordered product, 2 ∂

∂t2

+ω

2

#

ˆ q(t ˆ )] = T T [q(t)

$

¨ˆ + ω 2 q(t) ˙ˆ q(t)] ˆ ˆ ˆ ) + δ(t − t )[q(t), q(t) q(t

= 0 − iδ(t − t ) .

(A.1.26)

335

A.1 Bosonic ﬁelds Euclidean path integrals In Schr¨ odinger language, the time-ordered product (A.1.17) is ˆ ˆ > − t< )] qˆ exp[−iHt ˆ < ]|qi ,

qf | exp[−iH(T − t> )] qˆ exp[−iH(t

(A.1.27)

where t> is the greater of t and t , and t< is the lesser. By the nature of the path integral, which builds amplitudes from successive time steps, the time diﬀerences multiplying H are always positive in amplitudes such as (A.1.27): one only encounters exp(−iH∆t) for positive ∆t. Now continue analytically by uniformly rotating all times in the complex plane, t → e−iθ t for any phase 0 ≤ θ ≤ π. The real part of the exponent is −E sin θ∆t, where E is the energy of the intermediate state. Because the Hamiltonian is bounded below for a stable system, the sum over intermediate states is safely convergent and no singularities are encountered — the continued amplitude is well deﬁned. Taking the times to be purely imaginary, t = −iu for real u, deﬁnes the Euclidean amplitudes, as opposed to the Minkowski amplitudes studied above:

qf , u|T

ˆ a ) |qi , 0 E q(u

(A.1.28)

a

is deﬁned to be the continuation of

qf , t|T

ˆ a ) |qi , 0

q(t

(A.1.29)

a

to t = −iu. One can repeat the derivation of the path integral representation for the Euclidean case, but the result is extremely simple: just replace t in the action with −iu,

qf , U|T

ˆ a ) |qi , 0 E = q(u

a

U

qf ,U

[dq]q ,0 exp i

0

du L(q, i∂u q)

q(ua ) .

a

(A.1.30) The weight in the Euclidean path integral is conventionally deﬁned to be exp(−SE ), so the Euclidean action contains a minus sign,

SE =

du LE (q, ∂u q) = −

du L(q, i∂u q) .

(A.1.31)

Euclidean path integrals are usually the more well-deﬁned, because the exponential of the action is damped rather than oscillatory. The Minkowski matrix elements can then be obtained by analytic continuation. In the text our main concern is the Euclidean path integral, which we use to represent the string S-matrix. One is often interested in traces, such as ˆ exp(−Ei U) , (A.1.32) Tr exp(−HU) = i

336

Appendix A

ˆ and the Ei are the correwhere the sum runs over all eigenvectors of H sponding eigenvalues. In the q basis the trace becomes

ˆ dq q| exp(−HU)|q

=

=

dq q, U|q, 0 E

dq

q,U

[dq]q,0 exp(−SE )

[dq]P exp(−SE ) .

=

(A.1.33)

In the last line, [dq]P denotes the integral over all periodic paths on [0, U]. An interesting variation on this is the integral over all antiperiodic paths:

[dq]A exp(−SE ) =

=

=

q,U

[dq]−q,0 exp(−SE )

dq

dq q, U| −q, 0 E ˆ 0

dq q, U|R|q, E

ˆ Rˆ . = Tr exp(−HU)

(A.1.34)

Here Rˆ is the reﬂection operator, taking q to −q. We see that twisting the boundary conditions — for example, replacing periodic with antiperiodic — is equivalent to inserting an operator into the trace. Since Rˆ 2 = 1, its eigenvalues are ±1. If we take the average of the periodic and antiperiodic ˆ which projects onto the Rˆ = +1 space: path integrals, this inserts 12 (1 + R), 1 exp(−Ei U) . (periodic + antiperiodic) = 2 R =+1

(A.1.35)

i

In the same way, 12 (periodic − antiperiodic) gives the trace on the Rˆ = −1 space. ˆ is a Hermitean operator, A minor but sometimes confusing point: if A ˆ ˆ A ˆ exp(−Hu) ˆ satisﬁes the Euclidean Heisenberg operator A(u) = exp(Hu) ˆ ˆ † = A(−u) A(u)

(A.1.36)

and so is not in general Hermitean. For this reason it is useful to deﬁne a Euclidean adjoint, † ˆ ˆ . A(u) = A(−u)

(A.1.37)

A Hermitean operator in the Schr¨ odinger picture becomes Euclidean self-adjoint in the Euclidean Heisenberg picture.

337

A.1 Bosonic ﬁelds Diagrams and determinants

The diagrammatic expansion is easily obtained from the path integral. This subject is well treated in modern ﬁeld theory texts. We will only derive here a few results used in the text. For notational clarity we switch to ﬁeld theory, replacing q(t) with a real scalar ﬁeld φ(t, x). Begin with the free action 1 2

dd x φ(x)∆φ(x) ,

(A.1.38)

where ∆ is some diﬀerential operator. Consider the path integral with a source J(x), taking the Euclidean case to be speciﬁc,

.

d

exp i

Z[J] = =

d x J(x)φ(x)

1 [dφ] exp − 2

d

d

d x φ(x)∆φ(x) + i

d x J(x)φ(x) . (A.1.39)

This is the generating functional for Euclidean Green’s functions. Taking functional derivatives with respect to J(x) pulls down factors of iφ(x) into the integral. Shifting φ to φ + i∆−1 J completes the square, leaving

1 Z[J] = Z[0] exp − dd x dd y J(x)∆−1 (x, y)J(y) 2 1 δ2 d d −1 = Z[0] exp d x d y ∆ (x, y) 2 δφ(x)δφ(y)

× exp i

d x J(x)φ(x) d

. (A.1.40) φ=0

In the second line, each φ derivative becomes iJ. Any functional F of φ can be written F[φ] =

[dJ] exp i

dd x J(x)φ(x) f[J] ,

(A.1.41)

where f[J] is the (functional) Fourier transform of F. By linearity one obtains

1

F = Z[0] exp 2

δ2 d x d y ∆ (x, y) F[φ] . (A.1.42) δφ(x)δφ(y) φ=0 d

d

−1

Expanding the exponential generates the diagrammatic expansion, all ways of contracting pairs of ﬁelds in F. Also, when F is a product of factors, F1 F2 . . . , we can separate δ δ δ = + + ... , δφ(x) δφ1 (x) δφ2 (x)

(A.1.43)

338

Appendix A

where δ/δφi acts only on Fi . The self-contractions are then the terms δ2 δφi (x)δφi (y)

(no sum on i) .

(A.1.44)

Consider now the overall factor Z[0]. We assume that ∆ is self-adjoint and so we expand φ(x) in a complete set of eigenfunctions Φi (x): φ(x) =

φi Φi (x) ,

(A.1.45)

∆Φi (x) = λi Φi (x) ,

(A.1.46a)

i

where

dd x Φi (x)Φj (x) = δij . Deﬁne the path integral measure by [dφ] =

(A.1.46b)

(A.1.47)

dφi .

i

Then

1 =

i

=

1 dφi exp − φ2i λi 2

2π 1/2

λi

i

=

∆ det 2π

−1/2

.

(A.1.48)

The product of eigenvalues must be suitably regulated and renormalized. The value of 1 is usually given simply as (det ∆)−1/2 , because a rescaling of ∆ is equivalent to a rescaling of φ, the Jacobian for which can be absorbed into the bare couplings. (In free ﬁeld theory, the only ‘coupling’ that would be renormalized would be a c-number constant in the Lagrangian density; we will see this appearing in the example.) All of the above discussion extends readily to multicomponent q i or i φ . One case of interest is a complex scalar ﬁeld. One can break this up into two real scalars, φ = 2−1/2 (φ1 + iφ2 ), and the measure is [dφ dφ∗ ] ≡ [dφ1 dφ2 ]. For complex ﬁelds, (A.1.48) becomes

∗

[dφ dφ ] exp −

dd x φ∗ ∆φ = (det ∆)−1 .

(A.1.49)

One similarly ﬁnds for two real ﬁelds

[dφ1 dφ2 ] exp i

dd x φ1 ∆φ2 = (det ∆)−1 .

(A.1.50)

339

A.1 Bosonic ﬁelds This is a functional version of the delta-function integral ∞ −∞

dx

∞ −∞

dy exp(iλxy) = 2π

∞ −∞

dx δ(λx) =

2π . λ

(A.1.51)

An example As an example, we study the Euclidean transition amplitude for the harmonic oscillator,

qf , U|qi , 0 E =

qf ,U

[dq]q ,0 exp(−SE ) ,

(A.1.52a)

i

1 U

du (∂u q)2 + ω 2 q 2 + Sct , (A.1.52b) 2 0 where Sct is a cutoﬀ-dependent piece to be determined later. To evaluate this, separate out the ‘classical’ part of q,

SE =

q(u) = qcl (u) + q (u) ,

(A.1.53)

where −¨ qcl (u) + ω 2 qcl (u) = 0 , qcl (0) = qi , qcl (U) = qf .

(A.1.54a) (A.1.54b)

The boundary conditions on q are then q (0) = q (U) = 0

(A.1.55)

SE = Scl (qi , qf ) + S + Sct .

(A.1.56)

and the action separates

Here Scl (qi , qf ) =

1 2

U 0

du (∂u qcl )2 + ω 2 qcl2

(qi2 + qf2 ) cosh ωU − 2qi qf =ω , 2 sinh ωU S =

1 2

U 0

du (∂u q )2 + ω 2 q 2 .

The amplitude becomes

qf , U|qi , 0 = exp[−Scl (qi , qf ) − Sct ]

0,U

[dq ]0,0 exp(−S ) .

(A.1.57a) (A.1.57b)

(A.1.58)

Expand in a complete set of eigenfunctions of ∆ = −∂u2 + ω 2 subject to

340

Appendix A

the boundary condition (A.1.55),

2 1/2 jπu j 2π2 fj (u) = sin + ω2 , (A.1.59a) , λj = U U U2 ∞ 2 2 ∆ j π + ω2U 2 = det . (A.1.59b) 2 2π 2πU j=1 The inﬁnite product diverges. The Pauli–Villars regulator is a simple way to deﬁne it. Divide by the amplitude for a regulator oscillator of very high frequency Ω, det

∆ 2π

regulate

→

∞ 2 2 j π + ω2U 2 j=1

j 2 π 2 + Ω2 U 2

(A.1.60)

.

For j small, the terms in (A.1.60) are the same as those in (A.1.59b) up to a multiplicative constant, while for j large the product converges. The value of the product (A.1.60) is Ω sinh ωU . (A.1.61) ω sinh ΩU This follows from a standard inﬁnite product representation for the sinh. One can guess the result (A.1.61) by noting that it has the right poles and zeros in U, and the right value as U goes to zero. Keeping only terms that are nonvanishing as Ω → ∞,

1/2

1

qf , U|qi , 0 → exp −Scl (qi , qf ) + (ΩU − ln Ω) − Sct . 2 (A.1.62) To get a ﬁnite answer as Ω → ∞, we need ﬁrst to include a term 12 Ω in the Lagrangian Lct , canceling the linear divergence in (A.1.62). That is, there is a linearly divergent bare coupling for the operator ‘1’. It may seem strange that we need to renormalize in a quantum mechanics problem, but the power counting is completely uniform with quantum ﬁeld theory. The logarithmic divergence is a wavefunction renormalization. To determine the ﬁnite normalization, compare the path integral expression (A.1.62) to the canonical result as U → ∞. The U → ∞ amplitude is easily evaluated by inserting a complete set of states, since only the ground state |0

contributes in the limit: ω 2 sinh ωU

qf , U|qi , 0 → qf |0

0|qi exp(−E0 U)

ω 1/2 ω exp − (qf2 + qi2 + U) . (A.1.63) = π 2 On the other hand, the path integral expression (A.1.62) becomes ω 2 1 1/2 2

qf , U|qi , 0 → ω exp − (qf + qi + U) + (ΩU − ln Ω) − Sct . (A.1.64) 2 2

A.2 Fermionic ﬁelds Thus 1 Sct = 2 and

qf , U|qi , 0 =

U 0

du Ω −

ω 2π sinh ωU

341

1 Ω ln 2 π

(A.1.65)

exp(−Scl ) ,

(A.1.66)

1/2

which is the correct result. A.2

Fermionic ﬁelds

Consider the basic fermionic quantum system, two states | ↑ and | ↓ with ˆ raising operator χˆ and lowering operator ψ: ˆ ↓ = 0 , χˆ | ↓ = | ↑ , ψ| ˆ ↑ = | ↓ , χˆ | ↑ = 0 , ψ| ˆ χˆ } = 1 , ψˆ 2 = χˆ 2 = 0 . {ψ,

(A.2.1a) (A.2.1b) (A.2.1c)

Amplitudes will be expressed as path integrals over the classical analogs ˆ These are anticommuting c-numbers or Grassmann variables, of χˆ and ψ. elements χm and ψm of a Grassmann algebra {χm , ψn } = {χm , χn } = {ψm , ψn } = 0 .

(A.2.2)

There is a natural notion of integration over Grassmann variables, the Berezin integral. Notice that if θ is a Grassmann variable then θ2 = 0, and so the most general function of θ is of the form a + θb. Deﬁne

dθ = 0 ,

dθ θ = −

θ dθ = 1 .

(A.2.3)

Integrals of more general expressions are determined by requiring the integral to be linear and by deﬁning dθ to anticommute with all Grassmann variables. The signiﬁcance of the deﬁnition (A.2.3) is that the integral of a total derivative is zero: d dθ (a + θb) = dθ b = 0 . (A.2.4) dθ For multiple Grassmann variables the integral is

dθ1 . . . dθn f(θ) = c ,

(A.2.5)

where c is the coeﬃcient of the highest term in the Taylor expansion of f, f(θ) = . . . + θn θn−1 . . . θ1 c .

(A.2.6)

Looking back on the discussion of the bosonic path integral, one sees that all of the properties of ordinary integrals that were used are also true for

342

Appendix A

the Berezin integral: linearity, invariance (except for a sign) under change of order of integration, and integration by parts.3 In order to follow the bosonic discussion as closely as possible, it is ˆ useful to deﬁne states that are formally eigenstates of ψ: |ψ = | ↓ + | ↑ ψ , ˆ ψ|ψ

= |ψ ψ .

(A.2.7a) (A.2.7b)

ψ|ψ = ψ − ψ .

(A.2.8)

Deﬁne ψ| to satisfy The right-hand side is the Grassmann version of the Dirac delta function. It has the same properties as the ordinary delta function, (ψ − ψ )f(ψ) = (ψ − ψ )f(ψ ) and

dψ (ψ − ψ )f(ψ) = f(ψ )

(A.2.9)

(A.2.10)

for arbitrary f(ψ). The state ψ| is a left eigenstate with opposite sign,

ψ|ψˆ = −ψ ψ| ,

(A.2.11)

as one veriﬁes by taking the inner product of both sides with a general state |ψ . Note also that

|ψ dψ ψ| = 1 .

(A.2.12)

One can verify this by taking the matrix element between arbitrary states

ψ | and |ψ and evaluating the Grassmann integral. The delta function has an integral representation −

dχ exp[χ(ψ − ψ)] = ψ − ψ = ψ|ψ .

(A.2.13)

By expanding the inner product (A.2.8) in powers of ψ one obtains

ψ| ↓ = ψ ,

ψ| ↑ = −1 ,

(A.2.14)

and therefore

ψ|ˆχ|ψ = ψ| ↑

=−

3

dχ χ exp[χ(ψ − ψ)] .

(A.2.15)

It is not useful to think of dθ as an integral in any ordinary sense: it is simply a linear mapping from functions of θ to the complex numbers. Since it shares so many of the properties of ordinary integrals, the integral notation is a useful mnemonic.

343

A.2 Fermionic ﬁelds ˆ one obtains Using the further property that |ψ is an eigenstate of ψ, ˆ = − ψ|f(ˆχ, ψ)|ψ

dχ f(χ, ψ ) exp[χ(ψ − ψ)] .

(A.2.16)

This holds for any function f with the prescription that it is written with the ψˆ on the right. We can now emulate the bosonic discussion, using the completeness relation (A.2.12) to join together successive time steps, with the result

ψf , T |ψi , 0 ≈ −

dχ0 dψ1 dχ1 . . . dψN−1 dχN−1 × exp

→

N−1

−iH(χm , ψm ) − χm (ψm+1 − ψm )

m=0

! T

[dχdψ] exp 0

˙ − iH(χ, ψ) dt −χψ

"

.

(A.2.17)

The inconvenient sign in the integral representation has been oﬀset by reversing the order of Grassmann integrations. A typical fermionic Hamilˆ = mˆχψ. ˆ The corresponding Lagrangian is L = iχψ ˙ − H. For tonian is H such ﬁrst order Lagrangians the Hamiltonian and Lagrangian formulations coincide, and the result (A.2.17) is also the Lagrangian path integral. There are two useful inner products for this system, (A) : (B) :

↑| ↑ = ↓| ↓ = 1 ,

↑| ↓ = ↓| ↑ = 1 ,

↑| ↓ = ↓| ↑ = 0 ,

↑| ↑ = ↓| ↓ = 0 .

(A.2.18a) (A.2.18b)

Inner product (A) is positive deﬁnite and gives ψˆ † = χˆ . It is the inner product for an ordinary physical oscillator. Inner product (B) is not positive, and it makes ψˆ and χˆ Hermitean. It is the appropriate inner product for the Faddeev–Popov ghost system, as discussed in chapter 4. The derivation of the path integral was independent of the choice of inner product, but the relations (A.2.14) determine the expansion of ψ| in basis states in terms of the inner product. All of the discussion of the bosonic case applies to the fermionic path integral as well. The derivation is readily extended to multiple Fermi oscillators and to ﬁeld theory. Renormalization theory works in the same way as for the bosonic path integral. Fermionic path integrals again satisfy cutting relations, which simply amount to taking one of the iterated integrals (A.2.17) and saving it for the end. Insertion of ψ(t) and χ(t) in the path integral again gives the corresponding operator in the matrix element, and multiple insertions are again time-ordered. The path integral insertion ψ(t)ψ(t ) = −ψ(t )ψ(t)

(A.2.19)

344

Appendix A

becomes the operator ˆ ψ(t ˆ )] ≡ θ(t − t )ψ(t) ˆ ψ(t ˆ ) − θ(t − t)ψ(t ˆ )ψ(t) ˆ T[ψ(t) .

(A.2.20)

Note that a minus sign appears in the deﬁnition of time-ordering. The equations of motion and the Euclidean continuation work as in the bosonic ˆ case, but there is one important change. Consider a general operator A, ˆ = |j Aji , A|i

(A.2.21)

ˆ where i, j run over ↓ and ↑. Then by expanding A|ψ

in terms of the basis states one ﬁnds

ˆ dψ ψ|A|ψ

= A↓↓ − A↑↑

ˆ

ˆ , = Tr (−1)F A

(A.2.22)

where the fermion number Fˆ has been deﬁned to be even for | ↓ and odd for | ↑ . In particular,

ˆ

ˆ dψ ψ, U|ψ, 0 = Tr (−1)F exp(−HU) .

(A.2.23)

In contrast to the bosonic results (A.1.33) and (A.1.34), the periodic path ˆ integral gives the trace weighted by (−1)F and the antiperiodic path integral gives the simple trace. Feynman diagrams are as in the bosonic case, but the determinants work out diﬀerently. Consider

1 =

[dψ dχ] exp

d

d x χ∆ψ

,

(A.2.24)

where we have again switched to the notation of ﬁeld theory. Expand ψ in eigenfunctions of ∆, and χ in eigenfunctions of ∆T , ψ(x) =

ψi Ψi (x) ,

i

χ(x) =

χi Υi (x) ,

∆Ψi (x) = λi Ψi (x) ,

∆T Υi (x) = λi Υi (x) ,

dd x Υi (x)Ψj (x) = δij . Deﬁne [dψdχ] =

i dψi dχi .

1 =

(A.2.25a)

i

Then

(A.2.25b) (A.2.25c)

dψi dχi exp(λi χi ψi )

i

=

λi

i

= det ∆ .

(A.2.26)

345

Exercises Similarly, one ﬁnds for a single ﬁeld ψ that

[dψ] exp

dd x ψ∆ψ = (det ∆)1/2 ,

(A.2.27)

where ∆ must be antisymmetric. These fermionic path integrals are just the inverse of the corresponding bosonic integrals. For ﬁnite-dimensional integrals we have to be careful about 2πs, ∞

∞

1 dx dy exp(2πiλxy) = = λ −∞ −∞

−1

dψ

dχ exp(λχψ)

. (A.2.28)

Curiously, we can be more cavalier for functional determinants, where these can be eliminated by rescaling the ﬁelds. Bosonic path integrals actually give the absolute values of determinants, while fermionic path integrals give the determinant with a sign that depends on the precise order of Grassmann factors in the measure. We will ﬁx the sign by hand.4 If ∆ has zero eigenvalues, the path integral vanishes in the fermionic case and diverges in the bosonic case. When this occurs there is always a good reason, as we will see in practice, and the integrals of interest will have appropriate insertions to give a ﬁnite result. A useful result about Berezin integrals: from the deﬁnition it follows that if θ = θa then dθ = a−1 dθ. That is, the Jacobian is the inverse of the bosonic case. More generally, with bosonic variables xi and Grassmann variables θj , under a general change of variables to xi (x, θ), θj (x, θ) the inﬁnitesimal transformation of the measure has an extra minus sign, m

n

m

n

δ(d x d θ) = d x d θ

i

∂ ∂ δxi − δθj ∂xi ∂θj j

.

(A.2.29)

Exercises A.1 (a) Evaluate the path integral (A.1.33) for the harmonic oscillator by expanding [dq]P in a set of periodic eigenfunctions (not by inserting the result already obtained for the transition function into the left-hand side). Compare with an explicit evaluation of the sum over states (A.1.32). (b) Do the same for the antiperiodic path integral (A.1.34). A.2 Consider two harmonic oscillators of the same frequency, and form the complex coordinate q = 2−1/2 (q1 + iq2 ). Evaluate the following path 4

A ﬁne point: when the world-sheet theory is parity-asymmetric, even the phase of the path integral is ambiguous. The point is that ∆ is not Hermitian in this case, because the Euclidean adjoint includes a time-reversal (which is the same as a parity ﬂip plus Euclidean rotation). One must instead take the determinant of the Hermitian operator ∆† ∆, which determines that of ∆ only up to a phase. We will encounter this in section 10.7, for example, where we will have to check that there is a consistent deﬁnition of the phase.

346

Appendix A

integral on [0, U] by using a suitable mode expansion:

[dq dq ∗ ]θ exp(−SE ) ,

where the integral is to run over all paths such that q(U) = q(0)eiθ . Again compare with the answer obtained by summing over states: it is useful here to form complex combinations of oscillators, a1 ± ia2 . This is a more exotic version of exercise A.1, and arises in various string theories such as orbifolds. A.3 Consider a free scalar ﬁeld X in two dimensions, with action

1 d2 σ ∂1 X∂1 X + ∂2 X∂2 X + m2 X 2 . 4πα Evaluate the path integral with both coordinates periodic,

S=

0 ≤ σ1 ≤ 2π ,

0 ≤ σ2 ≤ T .

[Expanding the spatial dependence in Fourier modes, the answer takes the form of an inﬁnite product of harmonic oscillator traces.] A.4 Consider a free particle on a circle, H=−

1 ∂2 . 2 ∂φ2

By means of path integrals, evaluate φf , T |φi , 0 . Be careful: there are topologically inequivalent paths, distinguished by the number of times they wind around the circle. A.5 The fermionic version of exercise A.1: evaluate the fermionic path integral (A.2.23) and the corresponding antiperiodic path integral by using a complete set of periodic or antiperiodic functions, and compare with the sum over states.

References

The literature on the subjects covered in these two volumes easily exceeds 5000 papers, and probably approaches twice that number. To represent accurately the contributions of all who have worked in these many areas is an undertaking beyond the scope of this book. Rather, I have tried to assemble a list that is short enough to be useful in developing speciﬁc subjects and in giving points of entry to the literature. The resulting list includes many review articles, plus papers from the original literature where I felt they were needed to supplement the treatment in the text. Papers listed as ‘e-print: hep-th/yymmnnn’ are available electronically at the Los Alamos physics e-print archive, http://xxx.lanl.gov/abs/hep-th/yymmnnn . For a list of corrections and other useful information see http://www.itp.ucsb.edu/˜joep/bigbook.html . General references The reader may ﬁnd the two volume set by Green, Schwarz, & Witten (1987) (henceforth GSW) a useful supplement. The mix of covariant and light-cone methods used in that book, and the advanced topics emphasized, are somewhat complementary to the approach taken here. The book by Polyakov (1987) emphasizes conformal ﬁeld theory and the path integral approach to strings, but the focus is less on uniﬁed string theories and more on string theories of the strong interaction and on the statistical mechanics of random surfaces. Some other lectures on string theory are Peskin (1987), L¨ ust & Theisen (1989), Alvarez-Gaum´e & Vazquez-Mozo (1992), D’Hoker (1993), Ooguri & Yin (1997), and Kiritsis (1997). 347

348

References A brief history

There have been three periods of intense activity in the history of string theory, covering roughly the years 1968–73, 1984–9, and 1995 to the present. The ﬁrst of these began with the Veneziano amplitude as a phenomenological proposal for the S-matrix of the strong interaction. This led to dual resonance models, where the Veneziano formula was generalized and systematized. Further discoveries were the connection of dual models to strings, the critical dimension (26), superconformal invariance, and the existence of strings with critical dimension 10. This ﬁrst period of activity was ended by the discovery of QCD as the correct theory of the strong interaction, but those who had worked on string theory were left with the belief that they had discovered a rich physical theory that should have some application in nature. During the succeeding ‘quiet’ period, a small group of theorists developed string theory as a uniﬁed theory of gravity and gauge interactions. String theories with spacetime supersymmetry were discovered during this period, and the argument made that superstrings were a ﬁnite theory of quantum gravity. Three discoveries in 1984 established that string theory was also a strong candidate for the uniﬁcation of the Standard Model. These discoveries were the existence of anomaly-free chiral theories in ten dimensions, the heterotic string, and Calabi–Yau compactifaction. The ensuing period of activity was largely devoted to working out their consequences. The main diﬃculty that remained, and still remains, was the enormous number of potential vacua and an insuﬃcient understanding of the dynamics that chooses among them. The latest period of activity is marked by new dynamical tools. Again, many important results date from the preceding quiet period, but their importance and their relation to one another took time to emerge. The key steps were the systematic analysis of the consequences of supersymmetry in gauge theories and then in string theory. This led to many new discoveries: weak–strong duality in string theory, the uniﬁcation of all string theories as limits of a single theory, a limit in which a new, eleventh, dimension appears, the signiﬁcance of D-branes, and new understanding of the quantum mechanics of black holes. The developments from 1984 on are covered in the literature and review articles below and in volume two, but the earlier period is somewhat neglected. Many of the key papers up to 1985 can be found in Schwarz (1985), which also includes summaries of the history. Chapters 1 and 2 of the reference section of GSW give a more detailed history of the dual model period, with extensive references. The review by Scherk (1975) also covers this period.

References

349

Chapter 1 There are now many treatments of the Standard Model; Quigg (1983) gives a survey and Cheng & Li (1991) a more detailed treatment, to name just two. Weinberg (1979) discusses the UV problem of quantum gravity in ﬁeld theory. References for the various unifying ideas will be given in the appropriate chapters. The classical motion of the string and light-cone quantization are discussed in Goddard, Goldstone, Rebbi, & Thorn (1973) and in the reviews by Rebbi (1974) and Scherk (1975). The ﬁrst two of these include a careful treatment of Lorentz invariance. The heuristic argument that we have given is based on Brink & Nielsen (1973). Weinberg (1964) discusses the couplings of massless vector and tensor particles. Chapter 2 In addition to the general references, Friedan, Martinec, & Shenker (1986) covers many of the subjects in this chapter. Belavin, Polyakov, & Zamolodchikov (1984) is a classic reference on CFT, but directed at the advanced topics to be treated in chapter 15. Ginsparg (1990) is a review of CFT; Gatto (1973) and Fradkin & Palchik (1978) treat conformal invariance in four-dimensional quantum ﬁeld theory; although this is much smaller than the two-dimensional symmetry, many of the same ideas apply. Chapter 3 Additional references on the Polyakov path integral are Polyakov (1981), Friedan (1982), and Alvarez (1983). The Faddeev–Popov procedure is treated in all modern ﬁeld theory textbooks. The basic steps follow the original reference Faddeev & Popov (1967). The book by Birrell & Davies (1982) discusses Weyl anomalies in various dimensions (though it uses the common terminology conformal anomaly instead). Gravitational anomalies are discussed in Alvarez-Gaum´e & Witten (1983). An LSZ-like derivation of the vertex operators in light-cone gauge is in Mandelstam (1973); see also chapter 11 of GSW (1987). For calculation of beta functions in general nonlinear sigma models see Alvarez-Gaum´e, Freedman, & Mukhi (1981) and Friedan (1985). A general reference on strings in background ﬁelds is Callan, Friedan, Martinec, & Perry (1985); reviews include Callan & Thorlacius (1989) and Tseytlin (1989). The linear dilaton theory is discussed as a CFT by Chodos & Thorn (1974) and as a string background by Myers (1987).

350

References Chapter 4

Classic proofs of the OCQ no-ghost theorem are Brower (1972) and Goddard & Thorn (1972). The basic reference on quantization with constraints is Dirac (1950); a review is Henneaux (1985). The BRST symmetry in gauge theory is discussed in modern ﬁeld theory texts. The structure of its cohomology is discussed in Kugo & Ojima (1979). The BRST operator for ﬁeld theory is derived in Kato & Ogawa (1983); our proof of the noghost theorem is loosely patterned on the one in that paper. Other proofs in the BRST context are Freeman & Olive (1986), Frenkel, Garland, & Zuckerman (1986), and Thorn (1987). Chapter 5 D’Hoker & Phong (1988) is an extensive review of the subjects in this and subsequent chapters. The measure on moduli space is obtained in Moore & Nelson (1986) and D’Hoker & Phong (1986); see also Alvarez (1983). The representation of the measure in terms of the transition functions is discussed in Martinec (1987) and Alvarez-Gaum´e, Gomez, Moore, & Vafa (1988). General properties of the amplitude are also discussed in Friedan, Martinec, & Shenker (1986) and Mansﬁeld (1987). Chapter 6 Some of the physics of the tree-level amplitudes is discussed in Veneziano (1974). The appearance of gauge and gravitational interactions in the longdistance limit is developed in Neveu & Scherk (1972), Yoneya (1974), and Scherk & Schwarz (1974). Constraints on open string gauge groups are discussed in Marcus & Sagnotti (1982) and Schwarz (1982). For more on operator methods see GSW and the dual model references Schwarz (1973) and Mandelstam (1974). The relation of closed to open string tree-level amplitudes is given in Kawai, Lewellen, & Tye (1986). Chapter 7 The two volumes by Mumford (1983) are a beautiful introduction to theta functions; the one-loop functions are discussed in the ﬁrst half of volume one. Constraints from modular invariance on general conformal ﬁeld theories are in Cardy (1986). The zero-point energy in quantum ﬁeld theory is discussed in Coleman & Weinberg (1973). The cosmological constant problem is reviewed in Weinberg (1989).

References

351

Cancellation of divergences for open superstring gauge group S O(32) is discussed in Green & Schwarz (1985). The analysis using the boundary state formalism is in Callan, Lovelace, Nappi, & Yost (1987) and Polchinski & Cai (1988). Chapter 8 Kaluza–Klein theory is reviewed in Duﬀ, Nilsson, & Pope (1986) and in the reprint volume Appelquist, Chodos, & Freund (1987). Giveon, Porrati, & Rabinovici (1994) review toroidal compactiﬁcation and T -duality. Polchinski (1997) reviews T -duality of open and unoriented strings, and Dbranes. Further references on toroidal compactiﬁcation are Narain (1986) and Narain, Sarmadi, & Witten (1987). Dixon, Harvey, Vafa, & Witten (1985) is an introduction to orbifolds. Twisted state vertex amplitudes are obtained in Dixon, Friedan, Martinec & Shenker (1987) and Hamidi & Vafa (1987). More extensive orbifold references are given with chapter 16. The space of c = 1 CFTs is described in Dijkgraaf, Verlinde, & Verlinde (1988) and Ginsparg (1988) and in the review Ginsparg (1990). Chapter 9 Unitarity of tree-level amplitudes is discussed in the dual model reviews Schwarz (1973) and Mandelstam (1974) (these discussions are in OCQ, whereas section 9.1 gives the BRST version of the argument). One-loop amplitudes were originally constructed by imposing unitarity on the treelevel amplitudes; see Scherk (1975) for a review. The moduli space of higher genus Riemann surfaces is reviewed in Martinec (1987) and D’Hoker & Phong (1988). Some more details of the Schottky construction can be found in Mandelstam (1974) and references therein. Sewing of CFTs has been discussed by many authors. The treatment here is based on Vafa (1987), Polchinski (1988), Sonoda (1988a,b,1989) and LeClair, Peskin, & Preitschopf (1989a,b). The extension to the open string is in Lewellen (1992). The graphical decomposition of moduli space and the construction of minimal area metrics are reviewed in Zwiebach (1993). A discussion of the combinatorics of unitarity in quantum ﬁeld perturbation theory can be found in ’t Hooft & Veltman (1973). The Fischler–Susskind mechanism is reviewed in Polchinski (1988). For a diﬀerent approach to string unitarity, based on showing the equivalence between the light-cone and Polyakov amplitudes, see D’Hoker & Giddings (1987). The light-cone formalism is discussed in Mandelstam (1973) and GSW, chapter 11.

352

References

The form of open string ﬁeld theory that we have described is from Witten (1986). String ﬁeld theory is reviewed by Siegel (1988), Thorn (1989), and Zwiebach (1993). Borel summability of string perturbation theory is discussed in Gross & Periwal (1988). The 2g! behavior is discussed in Shenker (1991). High energy hard scattering is examined in Gross & Mende (1988) and Gross (1988). Another approach to high energy scattering is Amati, Ciafaloni, & Veneziano (1989). The spreading of the boosted string is described by Susskind (1993). The Hagedorn transition has been a source of much speculation since the original work of Hagedorn (1965). The discussion here is based in part on Atick & Witten (1988). The review Polchinski (1996) covers many of the subjects in this chapter, and in particular gives an introduction to matrix models. Other matrix model reviews are Kazakov & Migdal (1988), Klebanov (1992), Martinec (1992), and an extensive review by Ginsparg & Moore (1993). References Alvarez, O. (1983). Theory of strings with boundaries. Nuclear Physics, B216, 125. Alvarez-Gaum´e, L., Freedman, D. Z., & Mukhi, S. (1981). The background ﬁeld method and the ultraviolet structure of the supersymmetric nonlinear sigma model. Annals of Physics, 134, 85. Alvarez-Gaum´e, L., Gomez, C., Moore, G., & Vafa, C. (1988). Strings in the operator formalism. Nuclear Physics, B303, 455. Alvarez-Gaum´e, L., & Vazquez-Mozo, M. A. (1992). Topics in string theory and quantum gravity. In Les Houches Summer School on Gravitation and Quantizations, eds. J. Zinn-Justin & B. Julia, pp. 481–636. Amsterdam: North-Holland. E-print hep-th/9212006. Alvarez-Gaum´e, L., & Witten, E. (1983). Gravitational anomalies. Nuclear Physics, B234, 269. Amati, D., Ciafaloni, M., & Veneziano, G. (1989). Can spacetime be probed below the string size? Physics Letters, B216, 41. Appelquist, T., Chodos, A., & Freund, P. G. O. (1987). Modern Kaluza–Klein Theories. Menlo Park: Addison Wesley. Atick, J. J., & Witten, E. (1988). The Hagedorn transition and the number of degrees of freedom of string theory. Nuclear Physics, B310, 291. Belavin, A. A., Polyakov, A. M., & Zamolodchikov, A. B. (1984). Inﬁnite conformal symmetry in two-dimensional quantum ﬁeld theory. Nuclear Physics, B241, 333. Birrell, N. D., & Davies, P. C. W. (1982). Quantum Fields in Curved Space. Cambridge: Cambridge University Press.

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Brink, L., & Nielsen, H. B. (1973). A simple physical interpretation of the critical dimension of spacetime in dual models. Physics Letters, B45, 332. Brower, R. C. (1972). Spectrum-generating algebra and no-ghost theorem for the dual model. Physical Review, D6, 1655. Buscher, T. (1987). A symmetry of the string background ﬁeld equations. Physics Letters, B194, 59. Buscher, T. (1988). Path integral derivation of quantum duality in nonlinear sigma models. Physics Letters, B201, 466. Callan, C. G., Friedan, D., Martinec, E. J., & Perry, M. J. (1985). Strings in background ﬁelds. Nuclear Physics, B262, 593. Callan, C. G., Lovelace, C., Nappi, C. R., & Yost, S. A. (1987). Adding holes and crosscaps to the superstring. Nuclear Physics, B293, 83. Callan, C. G., & Thorlacius, L. (1989). Sigma models and string theory. In Particles, Strings, and Supernovae, TASI 1988, eds. A. Jevicki & C.-I. Tan, pp. 795–878. Singapore: World Scientiﬁc. Cardy, J. L. (1986). Operator content of two-dimensional conformally invariant theories. Nuclear Physics, B270, 186. Cheng, T.-P., & Li, L.-F. (1991). Gauge Theory of Elementary Particle Physics. Oxford: Oxford University Press. Chodos, A., & Thorn, C. B. (1974). Making the massless string massive. Nuclear Physics, B72, 509. Coleman, S., & Weinberg, E. (1973). Radiative corrections as the origin of spontaneous symmetry breaking. Physical Review, D7, 1888. D’Hoker, E. (1993). TASI Lectures on critical string theory. In Recent Directions in Particle Theory, TASI 1992, eds. J. Harvey & J. Polchinski, pp. 1–98. Singapore: World Scientiﬁc. D’Hoker, E., & Giddings, S. B. (1987). Unitarity of the closed bosonic Polyakov string. Nuclear Physics, B291, 90. D’Hoker, E., & Phong, D. H. (1986). Multiloop amplitudes for the bosonic Polyakov string. Nuclear Physics, B269, 205. D’Hoker, E., & Phong, D. H. (1988). The geometry of string perturbation theory. Reviews of Modern Physics, 60, 917. Dijkgraaf, R., Verlinde, E., & Verlinde, H. (1988). c = 1 conformal ﬁeld theories on Riemann surfaces. Communications in Mathematical Physics, 115, 649. Dirac, P. A. M. (1950). Generalized Hamiltonian dynamics. Canadian Journal of Mathematics, 2, 129. Dixon, L., Friedan, D., Martinec, E., & Shenker, S. (1987). The conformal ﬁeld theory of orbifolds. Nuclear Physics, B282, 13. Dixon, L., Harvey, J. A., Vafa, C., & Witten, E. (1985). Strings on orbifolds. Nuclear Physics, B261, 678.

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Glossary

(0, 2) compactiﬁcation a generic heterotic string vacuum having d = 4, N = 1 supersymmetry. The world-sheet CFT has N = 2 right-moving superconformal invariance. (2, 0) theory in current usage, this refers to a family of nontrivial ﬁxed point theories with tensionless strings and d = 6 (2, 0) supersymmetry. These arise on coincident M5-branes and IIA NS5-branes, and on the IIB theory at an A–D–E singularity. (2, 2) compactiﬁcation one of a special subset of d = 4, N = 1 heterotic string vacua, which includes the Calabi–Yau compactiﬁcations. The world-sheet CFT has both right-moving and left-moving N = 2 superconformal invariance. In the type II string theories, these CFTs give vacua with d = 4, N = 2 supersymmetry. A–D–E singularity a singularity of a four-(real)-dimensional complex manifold, resulting from the collapse of one or more two-spheres to zero volume. The terminology A–D–E refers to the Dynkin diagrams of the simply-laced Lie algebras, which describe the intersection numbers of the collapsed spheres. Abelian diﬀerential a globally deﬁned holomorphic (1,0)-form on a Riemann surface. abstruse identity one of a set of quartic theta function identities due to Jacobi, it implies the degeneracy of bosons and fermions in GSO-projected string theories as required by supersymmetry. aﬃne Lie algebra see current algebra. anomaly the violation of a classical symmetry by quantum eﬀects. A gravitational anomaly is an anomaly in coordinate invariance. A global anomaly is an anomaly in a large symmetry transformation (one not continuously connected to the identity). anomaly polynomial a formal (d + 2)-form in d-dimensions, which encodes the gauge and gravitational anomalies.

359

360

Glossary

asymptotically locally Euclidean (ALE) space a space which at long distance approaches ﬂat Euclidean space identiﬁed under a discrete group. This is the geometry in the neighborhood of an orbifold ﬁxed point (or blown-up ﬁxed point). Atiyah–Drinfeld–Hitchin–Manin (ADHM) construction a method for the construction of all Yang–Mills ﬁeld conﬁgurations having self-dual ﬁeld strength. auxiliary ﬁeld a nonpropagating ﬁeld, one whose ﬁeld equation is algebraic rather than diﬀerential. In many supersymmetric theories, the transformations can be simpliﬁed by introducing such ﬁelds. axion a Goldstone boson associated with spontaneously broken PQ symmetry. The model-independent axion appears in every perturbative string theory, and is closely related to the graviton and dilaton. bc CFT a free CFT of anticommuting ﬁelds with an action of ﬁrst order in derivatives. There is a family of such CFTs, parameterized by the weight hb = 1 − hc . For hb = 2 this CFT describes the Faddeev–Popov ghosts associated with conformal invariance. βγ CFT a free CFT of commuting ﬁelds with an action of ﬁrst order in derivatives. There is a family of such CFTs, parameterized by the weight hβ = 1 − hγ . For hβ = 32 this CFT describes the Faddeev–Popov ghosts associated with superconformal invariance. Batalin–Vilkovisky formalism an extension of the BRST formalism, for quantizing more general theories with constraints. This has been useful in string ﬁeld theory. Becchi–Rouet–Stora–Tyutin (BRST) invariance a nilpotent symmetry of Faddeev–Popov gauge-ﬁxed theories, which encodes the information contained in the original gauge symmetry. Beltrami diﬀerential the derivative with respect to the moduli of the complex structure of a Riemann surface. Berezin integration a linear operation taking functions of Grassmann variables to complex numbers, with many of the key properties of ordinary integration. beta function 1. in quantum ﬁeld theory, the derivative of the eﬀective strength of an interaction with respect to length scale; 2. a special function involving a ratio of gamma functions, which appears in the Veneziano amplitude. Betti numbers the number of nontrivial p-forms in de Rham cohomology, denoted Bp . black hole entropy a quantity S proportional to the area of the horizon of a black hole, S = 2πA/κ2 . This has the properties of a thermodynamic entropy: it is nondecreasing in classical general relativity, and the sum of the black hole entropy and the ordinary entropy is nondecreasing even with the inclusion of Hawking radiation. To ﬁnd a statistical mechanical

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derivation of this entropy has been a major goal, partly realized in recent work. black hole evaporation the emission of thermal (Hawking) radiation by a black hole, due to pair production near the horizon. black hole information paradox a conﬂict between quantum mechanics and general relativity. Information falling into a black hole is lost and does not reappear when the black hole evaporates; this is inconsistent with ordinary quantum mechanical evolution. It apparently requires either a signiﬁcant modiﬁcation of quantum mechanics, or a signiﬁcant breakdown of the usual understanding of locality. black p-brane a p-dimensional extended object with an event horizon: a space that is translationally invariant in p directions and has a black hole geometry in the remaining directions. blow up to deform a singular manifold into a smooth manifold. Bogomolnyi–Prasad–Sommerﬁeld (BPS) state a state that is invariant under a nontrivial subalgebra of the full supersymmetry algebra. Such states always carry conserved charges, and the supersymmetry algebra determines the mass of the state exactly in terms of its charges. BPS states lie in smaller supersymmetry representations than non-BPS states, so-called short representations. When there are short representations of diﬀerent sizes, one also distinguishes ultrashort representations, which are the smallest possible (generally their dimension is the square root of the non-BPS dimension). Borel summation a method of deﬁning the sum of a divergent series. This has been used as a means of studying nonperturbative eﬀects in ﬁeld and string theories, but it should be understood that most nonperturbative eﬀects are not usefully studied in terms of the perturbation series. Born–Infeld action a generalization of the usual gauge ﬁeld action which is nonpolynomial in the gauge ﬁeld strength. This was originally proposed as a possible short-distance modiﬁcation of electromagnetism. It arises as the low energy eﬀective action of the gauge ﬁelds on D-branes. bosonization the exact equivalence of a theory of fermionic ﬁelds and a theory of bosonic ﬁelds, possible in two dimensions. The boson is a fermion– antifermion pair; the fermion is a coherent state of bosons. c-map a method for constructing the hypermultiplet moduli space of a type II string theory compactiﬁed on a Calabi–Yau three-fold from the vector multiplet moduli space of the other type II theory on the same three-fold. c-theorem the existence, in unitary CFTs in two dimensions, of a positive quantity c that is monotonically nonincreasing with increasing length scale, and which at ﬁxed points is stationary and equal to the central charge. This is a strong constraint on the global form of the renormalization group ﬂow; no simple analog seems to exist in higher dimensions. Also known as the Zamolodchikov c-theorem.

362

Glossary

C PT symmetry the combined operation of parity-reversal, time-reversal, and charge conjugation, which is a symmetry of all Lorentz-invariant local quantum ﬁeld theories. Calabi–Yau manifold a K¨ahler manifold with vanishing ﬁrst Chern class. A Calabi–Yau n-fold has 2n real = n complex coordinates. Yau’s theorem guarantees the existence of a Ricci-ﬂat metric of SU(n) holonomy. canceled propagator argument a general principle implying, under broad conditions, the vanishing of surface terms on the moduli space of Riemann surfaces and therefore the decoupling of unphysical states in string amplitudes. Such amplitudes are deﬁned by analytic continuation from a regime where the integrand falls rapidly at the boundary and the surface term is identically zero; its continuation is therefore also identically zero. Casimir energy a shift in the ground state energy of a quantum ﬁeld theory due to boundary conditions on the ﬁelds. center-of-mass mode the zeroth spatial Fourier component of a quantum ﬁeld. central charge an operator (which might be a constant) that appears on the right-hand side of a Lie algebra and commutes with all operators in the algebra. Prominent examples include the constant term in the Virasoro algebra and the charges appearing on the right-hand sides of many supersymmetry algebras. Chan–Paton degrees of freedom degrees of freedom localized at the endpoints of open strings. These are now interpreted as designating the D-brane on which the string ends. Chan–Paton factor the vertex operator factor for the state of the Chan–Paton degrees of freedom. Chern–Simons term a term in the action which involves p-form potentials as well as ﬁeld strengths. Such a term is gauge-invariant as a consequence of the Bianchi identity and/or the modiﬁcation of the p-form gauge transformation. These terms usually have a close connection to topology and to anomalies. chiral 1. acting in a parity asymmetric fashion; see chiral multiplet, chiral symmetry, chiral theory, chirality, extended chiral algebra; 2. invariant under part of the supersymmetry algebra; see chiral ﬁeld, chiral multiplet, chiral primary, chiral ring. chiral ﬁeld in supersymmetry, a local operator that is invariant under part of the algebra: the operator analog of a BPS state. chiral multiplet the multiplet of d = 4, N = 1 supersymmetry with two real scalars. The quarks and leptons are contained in such multiplets. This multiplet is connected to both senses of chiral: it contains a fermion with chiral couplings, and the integral of the associated superﬁeld is invariant under half of the supersymmetry algebra. chiral primary in an N = 2 SCFT, a primary ﬁeld that is also annihilated by one of the rigid supersymmetries G± −1/2 .

Glossary

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chiral ring the closed OPE algebra of chiral ﬁelds. chiral symmetry a symmetry whose action on spinor ﬁelds is not paritysymmetric. chiral theory a gauge theory in which the gauge couplings are not paritysymmetric. chirality in d = 2k dimensions, the eigenvalue of the operator Γd which anticommutes with the Γµ . This eigenvalue distinguishes the two Weyl representations, which are related to one another by parity. Christoﬀel connection in general relativity, the connection that is constructed from the metric. critical behavior the behavior of a quantum ﬁeld theory at an IR ﬁxed point with massless ﬁelds, and the approach to this behavior. closed string a string with the topology of a circle. cocycle in a vertex operator, an operator-valued phase factor which multiplies the creation–annihilation normal ordered exponential. This in needed in some cases in order to give the operator the correct commuting or anticommuting property. coeﬃcient functions the position-dependent coeﬃcients of local operators appearing in the expansion of an operator product. cohomology in any vector space with a nilpotent operator Q (one such that Q2 = 0), the kernel of Q modulo the image of Q. That is, the space of closed states (those annihilated by Q) with the exact states (those of the form Qψ) deﬁned to be equivalent to zero. De Rham cohomology is the cohomology of the exterior derivative d acting on diﬀerential forms. On a complex manifold, Dolbeault cohomology is the cohomology of ∂ and ∂¯ (the (1, 0) and (0, 1) parts of d) on (p, q)-forms. Homology is the cohomology of the boundary operator. BRST cohomology is the cohomology of the BRST operator, and deﬁnes the physical space of a gauge-invariant theory. Coleman–Weinberg formula the expression for the vacuum energy density of a free quantum ﬁeld, from the renormalized sum of the zero-point energies of its modes. collapsing cycle a cycle whose volume vanishes in a limit, usually giving rise to a singular manifold. collective coordinate in quantizing a soliton or other extended object, the degrees of freedom corresponding to its position or conﬁguration. compact CFT a CFT in which the number of states with energy less than any given value is ﬁnite. This is deﬁned by analogy with the spectrum of a diﬀerential operator on a compact space. compactiﬁcation scale the characteristic mass scale of states whose wavefunctions have a nontrivial dependence on the compact dimensions. compactify to consider a ﬁeld theory or string theory in a spacetime, some of whose spatial dimensions are compact.

364

Glossary

complex manifold a manifold with an assigned system of complex coordinates, modulo holomorphic reparameterizations of these coordinates. complex structure an equivalence class of complex coordinates. A given diﬀerentiable manifold may have many inequivalent complex structures. complex structure moduli the moduli that parameterize the inequivalent complex structures on a manifold. In compactiﬁcation on a Calabi–Yau 3-fold, these are associated with (2, 1)-forms. conformal block in CFT, the contribution of a single conformal family to a sum over states. conformal bootstrap the partially successful program to construct all CFTs by using only symmetry and consistency conditions. conformal family the set of states obtained by acting on a highest weight state with Virasoro raising generators in all inequivalent ways; or, the corresponding set of local operators. A degenerate conformal family contains null states, which are orthogonal to all states in the family. conformal ﬁeld theory (CFT) a conformally invariant quantum ﬁeld theory. conformal gauge a choice of coordinates in two dimensions, such that the metric is proportional to the unit metric. conformal Killing vector a globally deﬁned inﬁnitesimal diﬀ×Weyl transformation that leaves the metric invariant. conformal transformation a mapping of Euclidean or Minkowski space to itself that leaves the ﬂat metric invariant up to a position-dependent rescaling; equivalently, the subgroup of diﬀ × Weyl that leaves invariant the ﬂat metric. In d ≥ 3 dimensions this has 12 (d + 1)(d + 2) parameters. In two dimensions it is the set of all holomorphic maps. Finite transformations require the inclusion of points at inﬁnity, as in the case of the M¨ obius transformations of the sphere. This usage has become standard in string theory and quantum ﬁeld theory, but in general relativity conformal transformation is deﬁned to be any position-dependent rescaling of the metric, now called a Weyl transformation in string theory. conifold a Calabi–Yau manifold with a singular complex structure, corresponding to the collapse of a three-cycle. The string theory on this space is singular; this is now understood to be due to the quantum eﬀects of a massless 3-brane wrapped on the cycle. conifold transition a change of topology due to condensation of massless 3brane ﬁelds. Under appropriate conditions, the potential for the massless 3-brane ﬁelds on a conifold with multiple collapsed cycles has a ﬂat direction for these ﬁelds; this corresponds to a change of topology, blowing up a 2-cycle rather than the collapsed 3-cycle. constraint a symmetry generator whose matrix elements are required to vanish in physical states, either in the BRST or OCQ sense. These can usually

Glossary

365

be understood as arising from a gauge symmetry (for so-called ﬁrst class constraints, which are all that we consider), and consist of those gauge symmetry generators that do not vanish by the equations of motion. coset CFT a CFT constructed as one of the factors of a known CFT, when the energy-momentum tensor of the latter can be written as a sum of commuting pieces. In the classic example the full CFT G is a current algebra, as is one of the factors (H). This can also be thought of as gauging the symmetry H. cosmological constant the energy density of the vacuum. In a nonsupersymmetric quantum theory (including one with spontaneously broken supersymmetry), there are many eﬀects that give rise to such an energy density. The cosmological constant problem is the problem that the cosmological constant in nature is many orders of magnitude smaller than known nonzero eﬀects. critical dimension the dimension in which a perturbative string theory is consistent in ﬂat Minkowski spacetime; the value is 26 for the bosonic string and 10 for the supersymmetric string theories. current algebra in quantum ﬁeld theory, the algebra of the currents associated with a continuous symmetry group g (or of their Fourier modes). As used here, it is the speciﬁc algebra that occurs in two-dimensional CFTs, with the energy-momentum tensor deﬁned to be of Sugawara form. The terms aﬃne Lie algebra and aﬃne Kac–Moody algebra are also used for this algebra, though like current algebra they both have broader deﬁnitions as well. The term aﬃne refers to the c-number (Schwinger) term. An untwisted current algebra is the algebra of periodic currents, with integer modes. An algebra can be twisted by any automorphism of g. cycle a topologically nontrivial submanifold (in the sense of homology); a pcycle is p-dimensional. The A- and B-cycles are a standard basis for the nontrivial one-cycles on a Riemann surface. D-brane in the type I, IIA, and IIB string theories, a dynamical object on which strings can end. The term is a contraction of Dirichlet brane. The coordinates of the attached strings satisfy Dirichlet boundary conditions in the directions normal to the brane and Neumann conditions in the directions tangent to the brane. A Dp-brane is p-dimensional, with p taking any even value in the IIA theory, any odd value in the IIB theory, and the values 1, 5, and 9 in the type I theory; a D9-brane ﬁlls space and so corresponds to an ordinary Neumann boundary condition. The Dp-brane is a source for the (p + 1)-form R–R gauge ﬁeld. The mass or tension of a D-brane is intermediate between that of an ordinary quantum or a fundamental string and that of a soliton. The low energy ﬂuctations of Dbranes are described by supersymmetric gauge theory, which is non-Abelian for coincident branes. D-instanton an object localized in (Euclidean) time as well as space, deﬁned by Dirichlet conditions on all coordinates of attached strings. This is similar to

366

Glossary a ﬁeld-theoretic instanton, corresponding to a tunneling process that changes the value of an R–R ﬁeld strength. More generally, the (p + 1)-dimensional world-volume of a Dp-brane, when localized in time and wrapped on a (p + 1)-cycle in space, has similar eﬀects.

D-string a D1-brane, in the type I and IIB string theories. D-term 1. in gauge theories with four or eight supersymmetries, the auxiliary ﬁeld in the gauge multiplet; 2 the potential term proportional to the square of this auxiliary ﬁeld, which depends only on the gauge couplings and, in the U(1) case, the value of the Fayet–Iliopoulos parameter. Del Guidice–Di Vecchia–Fubini (DDF) operators operators satisfying an oscillator algebra, which create a complete set of physical states in OCQ. descendant a state obtained by acting on a highest weight state with Virasoro raising generators. diagonal modular invariant a modular-invariant CFT formed by imposing common boundary conditions on the left- and right-moving ﬁelds. diﬀ invariance general coordinate (reparameterization) invariance, usually applied to the world-sheet coordinates. dilaton the massless scalar with gravitational-strength couplings, found in all perturbative string theories. An exactly massless dilaton would violate limits on nongravitational interactions, but a mass for the dilaton is not forbidden by any symmetry and so dynamical eﬀects will generate one in vacua with broken supersymmetry (the same holds for other moduli). The string coupling constant is determined by the value of the dilaton ﬁeld. dimensional reduction in the simplest cases, toroidal compactiﬁcation retaining only the states of zero compact momentum. More generally (and less physically) the construction of a lower-dimensional ﬁeld theory by requiring all ﬁelds to be invariant under a set of symmetries; this may have no interpretation in terms of compactiﬁcation. Dirac quantization condition for an electric and a magnetic charge, the condition that the product be quantized, µe µm = 2πn. For two dyons, which have both charges, the condition is µe µm − µm µe = 2πn. These conditions generalize to objects of dimension p and d − p − 4 in d dimensions, where one is the source of a (p + 2)-form ﬁeld strength and the other of the Poincar´e dual (d − p + 2)-form ﬁeld strength. Dirac spinor the unique irreducible representation of the algebra of Dirac matrices, which is also known as a Cliﬀord algebra. This is also a representation of the Lorentz group; in even dimensions it is reducible to two Weyl representations of the Lorentz group. The Dirac spinor is complex; in certain dimensions a further Majorana (reality) condition is compatible with Lorentz invariance. Dirichlet boundary condition the condition that the value of a ﬁeld be ﬁxed at a boundary. This is the relevant usage in string theory, but in other contexts only the tangent derivative need be ﬁxed.

Glossary

367

discrete torsion in forming a twisted CFT, a change in the phases of the path integral sectors and therefore in the projection on the Hilbert space. doubling trick the representation of holomorphic and antiholomorphic ﬁelds on a manifold with boundary by holomorphic ﬁelds alone, on the doubled copy of the manifold obtained by reﬂecting through the boundary. dual resonance model a phenomenological model of the strong interaction, which developed into string theory. The dual refers here to world-sheet duality. duality the equivalence of seemingly distinct physical systems. Such an equivalence often arises when a single quantum theory has distinct classical limits. One classic example is particle–wave duality, wherein a quantum ﬁeld theory has one limit described by classical ﬁeld theory and another described by classical particle mechanics. Another is the high-temperature– low-temperature duality of the Ising model. Here, low temperature is the statistical mechanical analog of the classical limit, the Boltzmann sum being dominated by the conﬁgurations of lowest energy. See Montonen–Olive duality, S-duality, string–string duality, T -duality, U-duality, world-sheet duality. eﬀective ﬁeld theory the description of a physical system below a given energy scale (or equivalently, above a given length scale). Einstein metric the metric whose leading low energy action is the Hilbert action 1 dd x (−G)1/2 R ; 2κ2 this is independent of other ﬁelds. Here κ is the gravitational coupling, related to the Planck length by κ = (8π)1/2 LP . This metric is related to other metrics such as the sigma-model metric by a ﬁeld-dependent Weyl transformation. The existence of distinct metrics would appear to violate the equivalence principle, but when the dilaton and other moduli are massive the distinction disappears. electroweak scale the mass scale of electroweak symmetry breaking, roughly 102 GeV. enhanced gauge symmetry a gauge symmetry appearing at special points in moduli space, which is not evident in the original formulation of a theory. The classic examples are the gauge symmetries that arise at special radii of toroidal compactiﬁcation, whose gauge bosons are winding states. Many other mechanisms are now known: D-branes and black p-branes wrapped on collapsing cycles, F-strings or D-strings stretched between various branes in the limit that the latter become coincident, and the gauge symmetry appearing on a zero size SO(32) instanton. Euclidean having a metric of strictly positive signature. The original connotation that the metric be ﬂat is somewhat disregarded; thus one refers to Euclidean quantum gravity, a conjectured analytic continuation of the Minkowskian theory. For the metric itself, the term Riemannian for a curved metric of Euclidean signature is more precise.

368

Glossary

Euclidean adjoint in a Euclidean quantum theory, the Hermitean adjoint combined with time-reversal. The latter operation undoes the time-reversing eﬀect of the adjoint, so that the combined operation is local. Euler number the topological invariant χ=

d

(−1)p Bp ,

p=0

where Bp is the pth Betti number. It is equal to 2(1 − g) for a Riemann surface of genus g. More properly, the Euler characteristic. expectation value a path integral with speciﬁed insertions. This is the term that we have chosen to use, but correlation function and correlator are also in common usage. extended chiral algebra the full set of holomorphic operators in a CFT. extended supersymmetry a supersymmetry algebra in which the supercharges comprise more than one copy of the smallest spinor representation for the given spacetime dimension. F -term 1. the auxiliary ﬁeld in the chiral multiplet of d = 4, N = 1 supersymmetry; 2. the potential term proportional to the square of this ﬁeld. F theory 1. a systematic description of IIB superstring states with nontrivial dilaton and R–R scalar backgrounds, which relates these ﬁelds to the modulus τ of an auxiliary two-torus; 2. a conjectured twelve-dimensional quantum theory underlying the IIB string. The name is inspired by Mtheory, with F for father. Faddeev–Popov determinant the Jacobian determinant arising from the reduction of a gauge-invariant functional integral to an integral over a gauge slice. Faddeev–Popov ghosts the wrong-statistics quantum ﬁelds used to give a functional integral representation of the Faddeev–Popov determinant. Fayet–Iliopoulos term in U(1) gauge theories with four or eight supersymmetries, a term in the action which is linear in the auxiliary D-term ﬁeld. Feigin–Fuchs representation a representation of minimal model expectation values in terms of free ﬁelds. ﬁbration a space which is locally the product of a ﬁber F and a base B. The geometry of the ﬁber varies as one moves over the base, and may become singular. Typical ﬁbers are tori and the K3 manifold. ﬁrst Chern class on a complex manifold, the Dolbeault cohomology class of the Ricci form Ri¯ dz i d¯z¯ . ﬁrst-quantized description the representation of a quantized particle theory as a sum over particle paths, or of a string theory as a sum over worldsheets. Second-quantized refers to the representation in terms of a functional integral over ordinary or string ﬁelds. The term second-quantized implies

Glossary

369

the reinterpretation of the ﬁrst-quantized wavefunction as a ﬁeld operator. This terminology is in common usage, but it has been argued that it is unsatisfactory, in that it implies a deep principle where none may exist. Since the sum over world-sheets is itself a quantum ﬁeld theory, one can equally well call it second-quantized, in which case string ﬁeld theory is thirdquantized. Third quantization of an ordinary ﬁeld theory would describe operators that create and destroy universes, a concept which may or may not be useful. Fischler–Susskind mechanism the cancellation of divergences and anomalies in the world-sheet quantum ﬁeld theory against divergences and anomalies from integration over small topological features at higher orders of string perturbation theory. This is needed for the consistency of string perturbation theory in a quantum-corrected background. ﬁxed point 1. (in geometry) a point left invariant by a given symmetry transformation. This becomes a boundary point or a singularity if the space is identiﬁed under the transformation; 2. (in quantum ﬁeld theory) a quantum theory whose physics is independent of length scale (scale-invariant). Usually such a theory is conformally invariant as well. A UV ﬁxed point is the theory governing the short-distance physics of a quantum ﬁeld theory; an IR ﬁxed point is the theory governing the long-distance physics of a quantum ﬁeld theory. A trivial IR ﬁxed point has no massless ﬁelds. A nontrivial IR ﬁxed point has massless ﬁelds with nonvanishing interactions. A theory whose IR limit is a massless free ﬁeld theory is therefore described by neither of these terms; it is a noninteracting IR ﬁxed point. ﬂat direction in scalar ﬁeld space, a line of degenerate local minima. The ﬁeld corresponding to this direction is a modulus. ﬂop a change of topology which can occur in weakly coupled string theory, where a two-cycle collapses and then a diﬀerent two-cycle blows up. fractional charge an unconﬁned particle whose electric charge is not a multiple of that of the electron. These exist in most d = 4 string theories, although in many cases all are superheavy. fractional string theory a proposed generalization of string theory having constraints whose spin is not a multiple of 12 . No complete construction exists. Fuchsian group a discrete subgroup Γ of the SL(2, R) M¨ obius transformations of the complex upper-half-plane H, with additional conditions such that H/Γ is a manifold and in particular a Riemann surface. functional integral in our usage, synonymous with path integral. fundamental region in relation to a coset space M/Γ where Γ is a discrete group, a region F such that every point in M is identiﬁed with exactly one point in the interior of F or with one or more points on the boundary of F. fundamental string (F-string) the original string whose quantization deﬁnes a weakly coupled string theory, as distinguished from D-strings and solitonic strings.

370

Glossary

fusion rule the speciﬁcation of which conformal families appear in the operator product of any two primary ﬁelds in a given CFT. gauge-ﬁxing the reduction of a redundant (gauge-invariant) description of a quantum theory to a description with a single representative from each equivalence class. gaugino a spin- 12 fermion in the same supersymmetry multiplet as a gauge boson. gaugino condensation a strong coupling eﬀect where a product of gaugino ﬁelds acquires a vacuum expectation value. This generally breaks a chiral symmetry but does not directly break supersymmetry; however, in combination with other ﬁelds it often induces supersymmetry breaking. generation a family of quarks and leptons, described by spinor ﬁelds in a chiral but anomaly-free set of SU(3) × SU(2) × U(1) representations. In SU(5) grand uniﬁcation these become a 5 + 10, in SO(10) they are contained in a 16, and in E6 they are contained in a 27. An antigeneration is the conjugate representation; the distinction between generation and antigeneration is a matter of convention. genus the number g of handles on a closed oriented Riemann surface: g = 0 is a sphere, g = 1 is a torus, and so on. Gepner model a string model based on N = 2 minimal model CFTs. ghosts see Faddeev–Popov ghosts. Gliozzi–Scherk–Olive (GSO) projection a construction of modular-invariant string theories by summing over R and NS boundary conditions on the fermion ﬁelds and projecting onto states of deﬁnite world-sheet fermion number. In supersymmetric string theories there are independent GSO projections on the left-movers and right-movers. The diagonal projection, which acts simultaneously on both sides, produces a nonsupersymmetric theory. goldstino the massless spin- 12 Goldstone fermion associated with spontaneously broken supersymmetry. In supergravity it combines with the gravitino to form a massive fermion. Goldstone boson the massless scalar corresponding to ﬂuctuations of the direction of spontaneous symmetry breaking. grand uniﬁcation the uniﬁcation of the SU(3)×SU(2)×U(1) gauge symmetries in a simple group. grand uniﬁcation scale the mass scale of spontaneous breaking of the grand uniﬁed group. Proton stability and the uniﬁcation of the couplings require that it be within two or three orders of magnitude of the gravitational scale. Grassmann variable the elements θi of an algebra with the relation θi θj = −θj θi . These are used to give a path integral representation of fermionic ﬁelds, and to deﬁne superspace. They are also called anticommuting cnumbers.

Glossary

371

gravitational scale the mass scale at which the dimensionless gravitational coupling becomes of order 1, mgrav = κ−1 = 2.4 × 1018 GeV; this is (8π)−1/2 times the Planck mass. gravitino a spin- 23 fermion in the same supersymmetry multiplet as the graviton. Green–Schwarz mechanism the cancellation of an anomaly by the modiﬁed transformation law of a p-form potential in a Chern–Simons term. Green–Schwarz superstring a manifestly supersymmetric formulation of the supersymmetric string theories, with a spacetime-fermionic gauge invariance known as κ symmetry. There is no simple covariant gauge ﬁxing. H-monopole a monopole carrying the magnetic charge of the antisymmetric tensor gauge ﬁeld Bµn . heterotic 5-brane the 5-brane carrying the magnetic charge of the massless heterotic string 2-form potential. It is obtained as the limit of a zero size instanton in the heterotic string gauge ﬁelds. The instanton conﬁguration is localized in four spatial dimensions, and is therefore a 5-brane in nine spatial dimensions. heterotic string a string with diﬀerent constraint algebras acting on the leftand right-moving ﬁelds. The case of phenomenological interest has a (0, 1) superconformal constraint algebra, with spacetime supersymmetry acting only on the right-movers and with gauge group E8 × E8 or SO(32). Hagedorn temperature the temperature at which the thermal partition function of free strings diverges, due to the exponential growth of the density of states of highly excited strings. hidden sector the ﬁelds that couple to the Standard Model only through gravitational-strength interactions. In hidden sector models, these include the ﬁelds responsible for supersymmetry breaking. highest weight state in CFT, a state annihilated by all Virasoro lowering operators, or more generally by all lowering operators in a given algebra. Hodge number the number of nontrivial (p, q)-forms in Dolbeault cohomology, denoted hp,q . holographic principle the conjecture that the states of quantum gravity in d dimensions have a natural description in terms of a (d − 1)-dimensional theory. This radical departure from local ﬁeld theory was motivated by the black hole information problem, and has played a role in attempts to formulate M-theory. holomorphic analytic, as used in the theory of complex variables. The Minkowskian continuation is left-moving. An antiholomorphic ﬁeld is analytic in the conjugate variable, and its continuation is right-moving. holomorphic quadratic diﬀerential a globally deﬁned holomorphic (2, 0)-form on a Riemann surface.

372

Glossary

holomorphic vector ﬁeld a globally deﬁned holomorphic (−1, 0)-form on a Riemann surface. holonomy consider the parallel transport of a vector around a closed loop on a d-dimensional manifold: it returns to an O(n) rotation of its original value. The set of all rotations that are obtained in this way for a given manifold is a subgroup of O(n); this is the holonomy group. homology see cohomology. hyperelliptic surface a Riemann surface with a Z2 symmetry. This can be represented as a two-sheeted cover of the sphere with branch cuts. hyper-K¨ahler manifold a 4k-dimensional manifold of holonomy Sp(k) ⊂ SO(4k). This is the geometry of the moduli space of hypermultiplets in d = 6, N = 1 or d = 4, N = 2 supersymmetry, in the limit in which gravity decouples. hypermultiplet in d = 6, N = 1 or d = 4, N = 2 supersymmetry, the multiplet whose bosonic content is four real massless scalars. identify to deﬁne two points (or other objects) to be equivalent, thus producing a coset space. infrared (IR) divergence a divergence arising from long distances in spacetime, usually signifying that one has calculated the wrong thing. inheritance principle in twisted (orbifold) theories, the principle that the treelevel amplitudes of untwisted states are the same as in the untwisted theory. insertion the integrand of a path integral, excluding the weight exp(iS) or exp(−S). instanton in a Euclidean path integral, a nonconstant conﬁguration that is a local but not a global minimum of the action. Such conﬁgurations are usually localized in spacetime, are usually topologically nontrivial, and are of interest when they give rise to eﬀects such as tunneling that are not obtained from small ﬂuctuations around a constant conﬁguration. Spacetime instantons are instantons in the eﬀective ﬁeld theory in spacetime. Worldsheet instantons are instantons in the world-sheet quantum ﬁeld theory, and correspond to world-sheets wrapping around nontrivial two-cycles of spacetime. intersection number the number of points at which a set of surfaces intersect, weighted by the orientation of the intersection. irrelevant interaction an interaction whose dimensionless strength decreases with increasing length scale. In perturbation theory, this is equivalent to a nonrenormalizable interaction. K3 manifold the unique nontrivial Calabi–Yau manifold of four (real) dimensions. To be precise, it is topologically unique, but possesses complex structure and K¨ ahler moduli. Its holonomy is SU(2), so that half of the supersymmetries of a theory are broken upon compactiﬁcation on K3.

Glossary

373

Kac determinant the determinant of the matrix of inner products of states at a given L0 level of a Verma module. Kac–Moody algebra see current algebra. K¨ahler form the (1, 1)-form Gi¯ dz i d¯z¯ , formed from a K¨ahler metric on a complex manifold. K¨ahler manifold a complex manifold of U(n) holonomy in n complex dimensions. K¨ahler moduli the moduli parameterizing the K¨ahler form. K¨ahler potential the potential K(z, ¯z ), in terms of which the metric of a K¨ahler manifold is determined, Gi¯ = ∂i ∂¯ K. This is not globally deﬁned, being determined only up to a K¨ahler transformation K(z, ¯z ) → K(z, ¯z ) + f(z) + f(z)∗ . Kaluza–Klein gauge ﬁeld in a compactiﬁed theory, a gauge ﬁeld originating from the metric of the higher-dimensional theory. The gauge group is the isometry group of the compact space. Kaluza–Klein monopole a monopole carrying the magnetic charge of a U(1) Kaluza–Klein gauge symmetry. The monopole conﬁguration is the smooth Taub–NUT spacetime. It is localized in three spatial dimensions, and is therefore a 6-brane in nine spatial dimensions. Kaluza–Klein states states with nonzero momentum in a compact spatial direction. Knizhnik–Zamolodchikov (KZ) equation the diﬀerential equation determining the expectation values of the primary ﬁelds of a current algebra. Landau–Ginzburg model a scalar ﬁeld theory which has a nontrivial IR ﬁxed point when the potential is appropriately tuned. In particular, this gives a Lagrangian representation of the minimal model CFTs. large coordinate transformation a coordinate transformation that is not continuously connected to the identity. lattice the set Γ of integer linear combinations of n linearly independent basis vectors in n dimensions. Given a Euclidean or Lorentzian metric, an even lattice is one whose points have even length-squared. The dual lattice Γ∗ is the set of points v such that v · w ∈ Z for all w ∈ Γ. The root lattice is the set of integer linear combinations of the roots of a Lie algebra. level 1. the quantized c-number term in a current algebra, also known as the Schwinger term; 2. the total oscillator excitation number in free ﬁeld theory; 3. in a conformal family, the diﬀerence between the L0 eigenvalue of a given state and that of the highest weight state. ˜ 0 ∈ Z. level-matching the modular invariance condition that L0 − L Lie algebra an algebra with an antisymmetric product that satisﬁes the Jacobi identity. A simple Lie algebra has no subalgebra that commutes with its complement. A simply-laced Lie algebra has all roots of equal length. A

374

Glossary graded Lie algebra has odd and even elements, with a symmetric product between odd elements.

light-cone gauge in string theory, the choice of world-sheet time coordinate to coincide with a particular spacetime null coordinate. In theories with local symmetries, a gauge choice such that the connection in a given null direction vanishes. linear dilaton theory a scalar CFT in which the energy-momentum tensor includes a term proportional to the second derivative of the scalar. This arises in string theory when the dilaton is a linear function of position. linear sigma model a scalar ﬁeld theory whose kinetic term is ﬁeld independent, but whose long-distance physics is governed by a nonlinear sigma model. Liouville ﬁeld theory the CFT of a scalar ﬁeld with an exponential interaction. This arises in various situations, including the noncritical string. It corresponds to bosonic string theory in a linear dilaton plus exponential tachyon background. little string theory one of several interacting string theories without gravity, notably found on NS5-branes in the limit of zero string coupling. loop expansion in quantum ﬁeld theory, the Feynman graph expansion, which is equivalent to the expansion in powers of ¯h. The string loop expansion is the sum over Riemann surfaces, with dimensionless string coupling g. The world-sheet loop expansion is the nonlinear sigma model perturbation expansion, in powers of α /Rc2 with Rc the compactiﬁcation radius. Lorentzian having a mixed signature (−, . . . , −, +, . . . , +). lowering operator operators that reduce the energy of a given state. In CFT, ˜ 0 ) by n units carry a operators that reduce the Virasoro generator L0 (or L grading (subscript) n. M-theory 1. (narrow) the limit of strongly coupled IIA theory with elevendimensional Poincar´e invariance; 2. (broad; most common usage) the entire quantum theory whose limits include the various weakly coupled string theories as well as M-theory in the narrow sense. The name is deliberately ambiguous, reﬂecting the unknown nature of the theory; M has variously been suggested to stand for membrane, matrix, mother, and mystery. M2-brane the 2-brane of M-theory, which couples to the potential A3 of eleven-dimensional supergravity. M5-brane the 5-brane of M-theory, which carries the magnetic charge of the potential A3 of eleven-dimensional supergravity. macroscopic string a string whose length is much greater than the characteristic string length scale. In particular, it is sometimes useful to consider an inﬁnite string stretching across spacetime. Majorana condition a Lorentz-invariant reality condition on a spinor ﬁeld. This can be imposed only if the spacetime dimension is 1, 2, 3, 4, or 8 (mod 8).

Glossary

375

marginal interaction an interaction whose dimensionless strength is independent of the length scale. In general this might hold only to ﬁrst order in the coupling of the interaction; a truly marginal interaction is one that remains marginal even with ﬁnite coupling. matrix models quantum mechanical systems with matrix degrees of freedom, with critical points governed by noncritical string theories. matrix theory a quantum mechanical system with matrix degrees of freedom and 32 supercharges, obtained by dimensional reduction of d = 10 supersymmetric U(n) Yang–Mills theory. In the large-n limit this is conjectured to deﬁne M-theory (in the broad sense). minimal models several families of solvable CFTs, in which every conformal family is degenerate. There are inﬁnite series of unitary minimal models having N = 0, N = 1, and N = 2 superconformal symmetries, which converge from below on the central charges 1, 32 , and 3 respectively. Minkowskian having a signature (−, +, +, . . . , +). mirror symmetry an equivalence between string theories compactiﬁed on distinct manifolds. The equivalence reverses the sign of the U(1) charge of one N = 2 superconformal algebra, and therefore changes the sign of the Euler number of the manifold. M¨obius group the globally deﬁned SL(2, C) conformal symmetry of the sphere; or, the globally deﬁned SL(2, R) conformal symmetry of the disk. mode operators the spatial Fourier components of a quantum ﬁeld. model see vacuum. modular group the group of large coordinate transformations (often applied to a Riemann surface but also applicable to spacetime). modular invariance the invariance of the string path integral under large coordinate transformations. moduli 1. the parameters labeling the geometry of a manifold. Notable examples are the parameters for the complex structure of the string world-sheet, and the parameters for the geometry of compactiﬁcation; 2. the parameters labeling a space of degenerate (and, usually, physically inequivalent) vacua in quantum ﬁeld theory. This is closely related to the compactiﬁcation example: in expanding around the classical limit, each compact solution of the ﬁeld equations gives a vacuum of the quantum theory, to leading order; 3. the massless ﬁelds corresponding to position dependence of these parameters. Contrast Goldstone boson. moduli space the space of geometries or vacua, whose coordinates are the moduli. monodromy for a quantity which is locally single-valued, the multi-valuedness around nontrivial closed paths. Montonen–Olive duality the weak–strong duality of d = 4, N = 4 Yang–Mills theory.

376

Glossary

Nambu–Goto action a string action, which is proportional to the invariant area of the world-sheet in spacetime. Narain compactiﬁcation the abstract description of toroidal compactiﬁcation in terms of the lattice of left- and right-moving momenta. naturalness problem the problem of explaining why a constant of nature takes a value much smaller than estimated nonzero contributions. Examples are the Higgs scalar mass, the cosmological constant, and the QCD θ-angle. Neumann boundary condition the condition that the normal derivative of a ﬁeld vanish at a boundary; the value of the ﬁeld is free to ﬂuctuate. Neveu–Schwarz algebra the world-sheet algebra of the Fourier modes of the supercurrent and energy-momentum tensor, in a sector where the supercurrent is antiperiodic and its moding therefore half-integer-valued. Neveu–Schwarz (NS) boundary condition the condition that a fermionic ﬁeld on the world-sheet be antiperiodic, in the closed string or in the double of the open string (see doubling trick). Its Fourier moding is then half-integervalued. Neveu–Schwarz 5-brane in the type I and type II superstring theories, the 5-brane that carries the magnetic charge of the NS–NS 2-form potential. Neveu–Schwarz–Neveu–Schwarz (NS–NS) states in type II super-string theories, the bosonic closed string states whose left- and right-moving parts are bosonic. These include the graviton and dilaton, and in the type II case a 2-form potential. no-ghost theorem 1. the theorem that the OCQ or BRST Hilbert space has a positive inner product; 2. the further theorem that the string amplitudes are well deﬁned and unitary in this space. no-scale model a ﬁeld theory that has, to some approximation, a line of degenerate vacua with broken supersymmetry. Noether’s theorem the theorem that an invariance of the Lagrangian implies a conserved quantity. noncommutative geometry a generalization of ordinary geometry, focusing on the algebra of functions on a space. The noncommutative collective coordinates of D-branes suggest the need for such a generalization. noncritical string theory 1. a Weyl-noninvariant string theory — one with a measure of world-sheet distance that is independent of the embedding in spacetime. These include strings with an independent world-sheet metric ﬁeld, and strings with a short-distance cutoﬀ; 2. more recently, the term has been applied to any string theory that does not have a weakly coupled limit with a Weyl-invariant world-sheet theory. In this form it includes various theories with stringlike excitations in which the coupling is ﬁxed to be of order 1, such as the (2,0) theory and the little string theories. Such a theory does not have a well-deﬁned world-sheet, because processes that change the world-sheet topology cannot be turned oﬀ.

Glossary

377

nonlinear sigma model a scalar ﬁeld theory in which the kinetic term has a ﬁeld-dependent coeﬃcient. This has a natural interpretation in terms of a curved ﬁeld space, and corresponds to string theory in curved spacetime, or more generally one with position-dependent background ﬁelds. nonrenormalization theorem a theorem restricting the form of quantum corrections to a given amplitude, or to the eﬀective action. It may require that these corrections vanish, that they arise only at speciﬁc orders of perturbation theory, or that they arise only nonperturbatively. normal ordering a prescription for deﬁning products of free ﬁelds by speciﬁc subtractions of divergent terms. Conformal normal ordering, denoted : :, produces operators with simple conformal properties. Creation–annihilation normal ordering, denoted ◦◦ ◦◦ , where lowering operators are put to the right of raising operators, produces operators with simple matrix elements. Boundary normal ordering, denoted , is conformal normal ordering with an additional image charge subtraction to produce operators that are ﬁnite as they approach a boundary. null state a physical state that is orthogonal to all physical states including itself. Or, a descendant in a conformal family which is orthogonal to all states in the family. old covariant quantization (OCQ) a method of quantizing string theory, similar to the Gupta–Bleuler quantization of electrodynamics. It is equivalent to the light-cone and BRST quantizations. one-loop the leading quantum correction, coming from surfaces of Euler number zero in string perturbation theory. open string a string that is topologically a line segment. operator equation in quantum theory, an equality between operators that holds in arbitrary matrix elements; equivalently, an equality that holds when inserted into a functional integral with arbitrary boundary conditions. operator product expansion (OPE) the expansion of a product of operators as a sum of local operators. This provides an asymptotic expansion, as the separation of the operators vanishes, for an arbitrary expectation value containing the product. In CFT the expansion is convergent. orbifold 1. (noun) a coset space M/H, where H is a group of discrete symmetries of a manifold M. The coset is singular at the ﬁxed points of H; 2. (noun) the CFT or string theory produced by the gauging of a discrete world-sheet symmetry group H. If the elements of H are spacetime symmetries, the result is a theory of strings propagating on the coset space M/H. A nonAbelian orbifold is one whose point group is non-Abelian. An asymmetric orbifold is one where H does not have a spacetime interpretation and which in general acts diﬀerently on the right-movers and left-movers of the string; 3. (verb) to produce such a CFT or string theory by gauging H; this is synonymous with the third deﬁnition of twist.

378

Glossary

oriented string theory a string theory in which the world-sheet has a deﬁnite orientation; world-sheet parity-reversal is not treated as a gauge symmetry. orientifold a string theory produced by the gauging of a world-sheet symmetry group H, where H includes elements that combine the world-sheet parityreversal Ω with other symmetries. orientifold plane a plane (of any dimension p) consisting of ﬁxed points of the orientifold group H (speciﬁcally, of an element of H that includes Ω). p-brane a p-dimensional spatially extended object. Examples are black p-branes, Dp-branes, M2- and M5-branes, NS5-branes, heterotic 5-branes, and (in d > 4) Kaluza–Klein monopoles. p-form a fully antisymmetric p-index tensor, usually written in an index-free notation. p-form gauge ﬁeld a generalization of Abelian gauge theory, with a p-form potential A, a (p − 1)-form gauge parameter λ, and a (p + 1)-form ﬁeld strength F. For p = 0 this is an ordinary massless scalar; for p = 1 it is an Abelian gauge ﬁeld. A (p + 1)-form potential couples naturally to a p-brane, through the integral of the form over the world-volume. A self-dual p-form theory is one where ∗F = F; this requires the spacetime dimension to be d = 2p + 2, and further d must be 2 mod 4 (in the Minkowskian case) in order that ∗∗ = 1. p-p string an open string with one endpoint on a Dp-brane and the other on a Dp -brane. (p, q)-form on a complex manifold, a tensor that is completely antisymmetric in p holomorphic indices and q antiholomorphic indices. (p, q) string a bound state of p F-strings and q D-strings in the IIB theory. parafermion CFT a family of coset CFTs with Zn symmetry, generalizing the Z2 -invariant free fermion theory. These describe the generic critical behavior of a system with Zn symmetry. parity transformation an operation that reﬂects one spatial dimension, or any odd number. One distinguishes spacetime parity, P (or β) and world-sheet parity Ω. partition function a sum over the spectrum of a quantum system, weighted by e−H/T where H is the Hamiltonian and T the temperature. Often additional charges are included in the exponent. This is the basic object in equilibrium statistical mechanics. In string theory it is given by a path integral on the torus or the cylinder, and so arises in one-loop amplitudes. path integral a representation of the transition amplitudes of a quantum system as a coherent sum over all possible histories. In quantum mechanics the history is a particle path; in quantum ﬁeld theory it is a path in ﬁeld space; in ﬁrst-quantized string theory it is the embedding of the string world-sheet in spacetime.

Glossary

379

Pauli–Villars regulator a means of regulating quantum ﬁeld theories by introducing a very massive wrong-statistics ﬁeld. Peccei–Quinn (PQ) symmetry an approximate symmetry, violated only by anomalies. period matrix a g × g matrix characterizing the complex structure of a genus-g Riemann surface. perturbation theory the expansion of the amplitudes of a quantum system in powers of the coupling. physical state in a quantum system with constraints, a state annihilated by the constraints. In OCQ this is a state annihilated by the Virasoro lowering generators and having a speciﬁed L0 eigenvalue.. In BRST quantization it is a state annihilated by the BRST operator. In both these cases, the true physical spectrum is the space of physical states with an additional equivalence relation, physical states diﬀering by a null state being identiﬁed. picture in the RNS superstring, one of several isomorphic representations of the vertex operators. The q-picture consists of vertex operators of βγ ghost charge q. The natural pictures are q = −1 and − 12 , with higher pictures including partial integrations over supermoduli space. The picture changing operator increases q by one. Planck length the natural length scale of quantum gravity, LP = MP−1 = 1.6 × 10−33 cm, constructed from ¯h, c, and GN . Planck mass the natural mass scale of quantum gravity, MP = 1.22 × 1019 GeV, constructed from ¯h, c, and GN . plumbing ﬁxture a procedure for constructing higher genus Riemann surfaces from lower ones by sewing in a handle. The construction includes a parameter q, such that when q goes to 0 the handle degenerates or pinches. This gives a canonical representation of the boundary of the moduli space of Riemann surfaces. Poincar´e dual a map from p-forms to (d − p)-forms, given by contraction with the completely antisymmetric tensor. Poincar´e invariance the invariance group of the ﬂat metric, consisting of translations and Lorentz transformations. point group the orbifold group H, with translations ignored (applicable only for orbifolds having a spacetime interpretation). Polyakov path integral a representation of ﬁrst-quantized string theory as a path integral with an independent world-sheet metric. A local Weyl symmetry guarantees that the classical degrees of freedom are the same as those of the Nambu–Goto theory. primary ﬁeld in CFT, a local operator annihilated by all of the lowering generators of a given algebra, such as the Virasoro algebra. The corresponding state is a highest weight state.

380

Glossary

projective space a compact n-dimensional space constructed from a linear (n + 1)-dimensional space by identifying points under the overall rescaling (x1 , . . . , xn+1 ) ∼ = (λx1 , . . . , λxn+1 ). For xi and λ real this produces RP n , and for xi and λ complex it produces CP n (which has n complex = 2n real dimensions). pseudospin in a current algebra, an SU(2) subalgebra not contained in the Lie algebra of the center-of-mass modes. puncture a marked point on a Riemann surface, the position of a vertex operator. QCD string a reformulation of non-Abelian gauge theory as a string theory, conjectured to exist at least in the limit of a large number of colors. quaternionic manifold a 4k-dimensional manifold with the holonomy group Sp(k) × SU(2) ⊂ SO(4k), with speciﬁc SU(2) curvature. This is the geometry of the moduli space of hypermultiplets in d = 6, N = 1 or d = 4, N = 2 supergravity. R symmetry a symmetry that acts nontrivially on the supercurrent(s). raising operator operators that reduce the energy of a given state. In CFT, ˜ 0 ) by n units carry operators that increase the Virasoro generator L0 (or L a grading (subscript) −n. Ramond algebra the world-sheet algebra of the Fourier modes of the supercurrent and energy-momentum tensor, in a sector where the supercurrent is periodic and its moding therefore integer-valued. Ramond (R) boundary condition the condition that a fermionic ﬁeld on the world-sheet be periodic, in the closed string or in the double of the open string (see doubling trick). Its Fourier moding is then integer-valued. Ramond–Neveu–Schwarz (RNS) superstring the formulation of type I and II superstrings that has superconformal invariance but not manifest spacetime supersymmetry. The latter emerges after imposing the GSO projection on the string Hilbert space. Ramond–Ramond (R–R) states in type I and type II superstring theories, the bosonic closed string states whose left- and right-moving parts are fermionic. These include p-form potentials Cp , with p taking all odd values in the IIA string and all even values in the IIB string. rank 1. the maximal number of commuting generators of a Lie algebra; 2. the number of indices on a tensor. rational CFT a CFT with a ﬁnite number of primary ﬁelds under an extended chiral algebra, a generalization of the minimal models. Such CFTs are highly constrained. refermionization after bosonization, the construction of new spin- 12 ﬁelds from linear combinations of the bosonic ﬁelds. These fermions are nonlocal, nonlinear functions of the original fermions.

Glossary

381

Regge behavior in scattering at large center-of-mass energy-squared s and ﬁxed momentum transfer-squared −t, the scaling of the amplitude as sα(t) . The values of t where α(t) = j is a nonnegative integer correspond to exchange of a particle of spin j and mass-squared −t. Regge slope denoted α , the square of the characteristic length scale of perturbative string theory. The tension of the fundamental string is 1/2πα . relevant interaction an interaction whose dimensionless strength increases with distance. In perturbation theory, this is equivalent to a superrenormalizable interaction. renormalization group equation the diﬀerential equation governing the change of physics with length scale. renormalization theory the calculus of path integrals. Riemann–Roch theorem the theorem that the number of metric moduli minus the number of conformal Killing vectors on a Riemann surface is −3χ, with χ being the Euler number of the surface; and, generalizations of this result. Riemann surface a two-(real)-dimensional complex manifold, equivalent to a Weyl equivalence class of Riemannian manifolds. root a vector of the eigenvalues of the maximal set of commuting generators of a Lie algebra, associated with a state in the adjoint representation. S-duality a duality under which the coupling constant of a quantum theory changes nontrivially, including the case of weak–strong duality. Important examples are the SL(2, Z) self-dualities of IIB string theory and of d = 4, N = 4 supersymmetric Yang–Mills theory. More loosely, it is used for weak–strong dualities between diﬀerent theories, such as IIA–M-theory (on a circle) duality, SO(32) heterotic–type I duality, and E8 heterotic–M-theory (on an interval) duality. In compactiﬁed theories, the term S-duality is limited to those dualities that leave the radii invariant, up to an overall coupling-dependent rescaling; contrast T-duality and U-duality. S-matrix the overlap amplitude between states in the inﬁnite past and states in the inﬁnite future; the scattering amplitude. In coordinate-invariant quantum theories this is generally the simplest invariant. The term usually implies a basis of free particle states; this is problematic in theories with massless particles due to IR divergences, and meaningless in theories at nontrivial IR ﬁxed points. scale transformation a rigid rescaling of spacetime, or of the world-sheet. Scherk–Schwarz mechanism the breaking of supersymmetry by dimensional reduction that includes a spacetime rotation. Schottky group a discrete subgroup Γ of the SL(2, C) M¨ obius transformations of the sphere S2 , with additional conditions such that S2 /Γ is a manifold and in particular a Riemann surface. Schwarzian the combination of derivatives appearing in the ﬁnite conformal transformation of the energy-momentum tensor.

382

Glossary

Schwinger–Dyson equation the operator equations of a quantum theory, expressed as equations for the expectation values. short multiplet see BPS state. sigma model metric the metric appearing in the string world-sheet action. Also known as the string metric, this diﬀers from the Einstein metric by a dilatondependent Weyl transformation. simple current an operator J such that, for any primary ﬁeld O, the operator product JO contains only a single conformal family. soliton a state whose classical limit is a smooth, localized, and (usually) topologically nontrivial classical ﬁeld conﬁguration; this includes particle states, which are localized in all directions, as well as extended objects. By contrast, a state of ordinary quanta is represented near the classical limit by small ﬂuctuations around a constant conﬁguration. In a theory with multiple classical limits (dualities), solitons and quanta may exchange roles. special K¨ahler geometry the geometry of the moduli space of vector multiplets in d = 4, N = 2 supergravity. It is most simply deﬁned (section B.7) in terms of special coordinates, which are ﬁxed up to a symplectic transformation. Rigid special geometry is obtained in the limit where gravity decouples and the supersymmetry becomes global. spectral ﬂow the adiabatic change in the spectrum produced by a continuous change in the boundary conditions. spin the behavior of a ﬁeld or a state under rotations. In a CFT this is given in terms of the conformal weights by h − ˜h. spin ﬁeld the vertex operator for a Ramond ground state, which produces a branch cut in the spinor ﬁelds. spin structure one of a set of inequivalent ways of deﬁning a spinor ﬁeld globally on a manifold. Roughly speaking, it corresponds to a choice of signs in the square roots of the transition functions. spurious state in OCQ, a state produced by Virasoro raising operators. state–operator isomorphism in CFT, a one-to-one isomorphism between states of the theory quantized on a circle and local operators. Also, a one-toone isomorphism between states of the theory quantized on an interval and local operators on a boundary. In d dimensions the circle becomes a (d − 1)-sphere and the interval a (d − 1)-hemisphere. string coupling the dimensionless parameter g governing the weights of diﬀerent Riemann surfaces in string perturbation theory, the contribution from surfaces of Euler number χ being weighted by g −χ . The string coupling is related to the dilaton by g = eΦ . This deﬁnition corresponds to the amplitude to emit a closed string; the amplitude to emit an open string is proportional to g 1/2 . string ﬁeld theory the representation of string theory as theory of ﬁelds, the ﬁelds being maps from a circle (or interval) into spacetime. This corresponds

Glossary

383

to an inﬁnite number of ordinary quantum ﬁelds. This formalism can reproduce string perturbation theory, but it is unclear whether it can be deﬁned beyond perturbation theory. string metric see sigma model metric. string scale the mass scale α−1/2 characterizing the tower of string excitations. string–string duality a term sometimes used to denote a weak–strong duality between diﬀerent string theories, in particular between the heterotic string compactiﬁed on T 4 and the IIA string compactiﬁed on K3. string tension the mass per unit length of a string at rest, related to the Regge slope by 1/2πα . Sugawara construction in current algebra, the construction of the energymomentum tensor as a product of two currents. Originally proposed as a phenomenological model in four dimensions, this was later found to be an exact result in two dimensions. superconformal algebra an extension of the conformal (Virasoro) algebra to in˜ superconformal algebra clude anticommuting spinor generators. The (N, N) ˜ has N left-moving and N right-moving supercurrents. superconformal current algebra an extension of the conformal transformations to include both spin- 23 and spin-1 currents. superconformal ﬁeld theory (SCFT) a quantum ﬁeld theory that is invariant under superconformal transformations. supercurrent a conserved spinor current. This includes the world-sheet current TF associated with superconformal transformations, and the spacetime current associated with spacetime supersymmetry. superﬁeld a ﬁeld on superspace, with speciﬁc transformation properties under a change of coordinates. supergravity the union of general relativity and supersymmetry, implying also the promotion of supersymmetry to a local symmetry. supermanifold (or superspace) a formal extension of the concept of manifold to include both commuting and anticommuting (Grassmann) coordinates. supermoduli the anticommuting parameters characterizing a super-Riemann surface. superpartner scale the mass scale of the superpartners of the Standard Model particles. This is expected to be between 102 and 103 GeV if supersymmetry solves the naturalness problem of the Higgs scalar mass. superpotential in d = 4, N = 1 supersymmetry, the holomorphic function of the superﬁelds that determines the nongauge interactions. super-Riemann surface a supermanifold deﬁned in terms of superconformal transition functions between patches. supersymmetry a symmetry whose charge transforms as a spinor, which relates the masses and couplings of fermions and bosons.

384

Glossary

supersymmetry breaking scale the mass scale of the expectation value that breaks supersymmetry. The superpartner scale is the supersymmetry breaking scale times the strength of the coupling of the Standard Model ﬁelds to the supersymmetry breaking ﬁelds. T -duality a duality in string theory, usually in a toroidally compactiﬁed theory, that leaves the coupling constant invariant up to a radius-dependent rescaling and therefore holds at each order of string perturbation theory. Most notable is R → α /R duality, which relates string theories compactiﬁed on large and small tori by interchanging winding and Kaluza–Klein states. More generally it includes shifts of antisymmetric tensor backgrounds and large coordinate transformations in spacetime. Contrast S-duality and U-duality. ’t Hooft–Polyakov monopole a classical solution with magnetic charge, which exists whenever a simple group is spontaneously broken to a group with a U(1) factor. tachyon a particle (almost always a scalar) with a negative mass-squared, signifying an instability of the vacuum. tadpole an amplitude for creation of a single particle from the vacuum, induced by quantum eﬀects. target space the space in which a function takes its values. This is usually applied to the nonlinear sigma model on the string world-sheet, where the target space is itself spacetime. Teichm¨uller parameters the moduli for the complex structure of a Riemann surface (strictly speaking, points in Teichm¨uller space are not identiﬁed under the modular group). tensionless string theory an interacting theory with tensionless strings. These can arise as p-branes with p − 1 directions wrapped on a collapsing cycle, as various 2-branes with one direction stretched between higher-dimensional branes when the latter become coincident, and on zero-size E8 × E8 instantons. In general the coupling is ﬁxed to be of order one, so there is no perturbation expansion. Tensionless implies that the string tension in units of the gravitational scale goes to zero; it is not applied to the fundamental string, which becomes noninteracting in that limit. tensor multiplet 1. the multiplet of d = 6, (1, 0) supersymmetry whose bosonic content is one self-dual tensor and one scalar. This reduces to a vector multiplet of d = 4, N = 2 supersymmetry; 2. the multiplet of d = 6, (2, 0) supersymmetry whose bosonic content is one self-dual tensor and ﬁve scalars. This reduces to a vector multiplet of d = 4, N = 4 supersymmetry. tensor operator in CFT, a local operator whose conformal or superconformal transformation involves only the ﬁrst derivative of the transformation; synonymous with a primary ﬁeld of the conformal or superconformal symmetry. Such an operator is mapped to a highest weight state by the state–operator isomorphism.

Glossary

385

tetrad a basis of d orthonormal vector ﬁelds in d dimensions. The term tetrad originates in d = 4 (as does the equivalent term vierbein) but there is no other convenient term for general d. the theory formerly known as strings see M-theory, second deﬁnition. theory see vacuum. theta functions the family of holomorphic functions having simple periodicity properties on a torus. Thirring model the solvable quantum ﬁeld theory of a single Dirac fermion with a quartic interaction in 1 + 1 dimensions. This is equivalent under bosonization to a free scalar ﬁeld at a general real radius. threshold correction a correction to the low energy eﬀective action, and in particular to the gauge coupling, due to virtual massive particles. topological string theory a modiﬁcation of string theory without local dynamics; all observables are topological. toric geometry a generalization of the idea of projective space. Roughly speaking, this corresponds to the most general linear sigma model. toroidal compactiﬁcation the periodic identiﬁcation of one or more ﬂat dimensions. torsion a term applied to various 3-form ﬁeld strengths, so called because they appear in covariant derivatives in combination with the Christoﬀel connection. tree-level the Feynman graphs which become disconnected if one propagator is cut, or the analogous string amplitudes, the sphere and disk. These correspond to classical terms in the eﬀective action. twist 1. (verb) to deﬁne a ﬁeld in a periodic space to be aperiodic by a symmetry transformation h; 2. (noun) the aperiodicity h; 3. (verb) given a CFT or string theory, to construct a new theory using a symmetry group H. One adds closed strings twisted by any of the elements h ∈ H, and requires all states to be invariant under the transformations in H. This is equivalent to treating H as a world-sheet gauge symmetry. The term orbifold is also used as a synonym; 4. (noun, archaic) world-sheet parity. twisted state a closed string with twisted periodicity. type ... supergravity the low energy supergravity theory of the corresponding string theory. The Roman numeral signiﬁes the number of d = 10 supersymmetries, and IIA and IIB distinguish whether the two supersymmetries have opposite or identical chiralities respectively. type I superstring the theory of open and closed unoriented superstrings, which is consistent only for the gauge group SO(32). The right-movers and leftmovers, being related by the open string boundary condition, transform under the same spacetime supersymmetry.

386

Glossary

type IIA superstring a theory of closed oriented superstrings. The right-movers and left-movers transform under separate spacetime supersymmetries, which have opposite chiralities. type IIB superstring a theory of closed oriented superstrings. The right-movers and left-movers transform under separate spacetime supersymmetries, which have the same chirality. type 0 string a pair of nonsupersymmetric string theories, which have tachyons and no spacetime fermions, with the same world-sheet action as the type II theories but with diﬀerent projections on the Hilbert space. U -duality any of the dualities of a string theory, usually of a toroidally compactiﬁed type II theory. This includes the S-dualities and T-dualities, but in contrast to these includes also transformations that mix the radii and couplings. ultrashort multiplet see BPS state. unit gauge a choice of coordinate and Weyl gauges in two dimensions, such that the metric is the identity. unitary as applied to a quantum system, the property of having a conserved inner product in a positive-norm Hilbert space. unoriented string theory a string theory in which world-sheet parity-reversal Ω is a discrete gauge symmetry. The perturbation theory includes unoriented world-sheets, and the spectrum is restricted to states with Ω = +1. UV divergence a divergence arising from short distances in spacetime, usually signifying a limit to the validity of a theory. vacuum a stable Poincar´e-invariant state. The novel feature of systems with unbroken supersymmetries is the frequent appearance of degenerate but physically inequivalent vacua. While it is now clear that the diﬀerent string theories are actually diﬀerent vacua in a single theory, it is still common to use the term theory for each. Also, diﬀerent CFTs within a single string theory are sometimes referred to as diﬀerent theories rather than vacua. The term model is used to refer to string vacua whose low energy physics resembles the Standard Model. vector multiplet in d = 4, N = 1 or d = 6, N = 1 supersymmetry, the multiplet whose bosonic content is a massless vector ﬁeld. The d = 6 multiplet reduces to a vector ﬁeld plus two real scalars in d = 4, N = 2 supersymmetry. Veneziano amplitude the bosonic string tree-level amplitude for four open string tachyons. Verma module the set of states obtained by acting on a highest weight state with Virasoro raising generators in all inequivalent ways, with the requirement that all such states be linearly independent (the generators act freely). This gives a representation of the Virasoro algebra that depends on the central charge c and on the weight h of the highest weight state. This is

Glossary

387

very similar to a conformal family, except that in the case of a degenerate representation some states in the latter may vanish; the Verma module would in this case give a reducible representation. vertex operator a local operator on the string world-sheet, corresponding to a string in the initial or ﬁnal state. Virasoro algebra in a CFT, the inﬁnite-dimensional Lie algebra of the Fourier modes of the energy-momentum tensor. Virasoro–Shapiro amplitude the bosonic string tree-level amplitude for four closed string tachyons. W algebras a family of extended chiral algebras with currents of spin greater than 2. W string a proposed generalization of string theory with constraints of spin greater than 2. No complete construction exists. Ward identity a relation between the divergence of the expectation value of a conserved current and the same expectation value without the current. ˜ 0 eigenvalue, which determines the behavior weight 1. in CFT, the L0 or L of an operator under scale transformations and rotations (= conformal weight); 2. in Lie algebra, a vector of the eigenvalues of the maximal set of commuting generators of a Lie algebra, in a state of a given representation. Wess–Zumino consistency condition the condition that the second derivative of a functional integral with respect to the background ﬁelds be symmetric. This strongly constrains the form of possible anomalies in a theory. Wess–Zumino–Novikov–Witten (WZNW) model a conformally invariant nonlinear sigma model on a group manifold with an antisymmetric tensor background. Weyl anomaly an anomaly in the Weyl transformation, this determines the critical dimension in the Polyakov formalism. This is sometimes called the trace anomaly or the conformal anomaly. Weyl condition in even dimensions, the condition that a fermion have deﬁnite chirality. This deﬁnes a spinor representation of the Lorentz group, which contains half of the components of the Dirac representation of the gamma matrix algebra. Weyl transformation a position-dependent rescaling of the metric. Wilson line 1. a gauge ﬁeld with vanishing ﬁeld strength but with nontrivial parallel transport (holonomy) around nontrivial paths in spacetime. 2. the gauge-invariant operator that measures such a ﬁeld: the trace of the pathordered product of the line integral of the vector potential. Wilsonian action the action in a low energy eﬀective ﬁeld theory, which incorporates the eﬀects of higher energy virtual states. winding state a closed string whose conﬁguration is a nontrivial path in a non-simply-connected spacetime.

388

Glossary

world-sheet 1. the two-dimensional surface in spacetime swept out by the motion of a string. 2. the abstract two-dimensional parameter space used to describe the motion of a string. world-sheet duality the equivalence between Hamiltonian descriptions obtained by cutting open a string world-sheet along inequivalent circles. Important examples are the associativity condition, from the four-point sphere amplitude, and the equivalence of the open string loop and closed string tree descriptions of the cylinder. wrapped refers to a p-brane, q of whose dimensions are wound on a nontrivial compact submanifold of spacetime, leaving a (p − q)-dimensional extended object. Yang–Mills ﬁeld non-Abelian gauge ﬁeld. Zamolodchikov metric the expectation value of a pair of local operators on the sphere, giving a natural inner product for the corresponding string states. This is the metric that appears in the kinetic term in the spacetime action. zero modes 1. of a diﬀerential operator: eigenfunctions with zero eigenvalue. In a Gaussian functional integral, these would give an inﬁnite factor in the bosonic case and a zero factor in the fermionic case. In general these have a physical origin and the functional integral has an appropriate integrand to give a ﬁnite result; 2. center-of-mass modes. zero-point energy the energy due to vacuum ﬂuctuations of quantum ﬁelds. zero-slope limit the limit α → 0. Only massless string states, described by low energy ﬁeld theory, remain.

Index Page numbers in italics refer to exercises

Abelian diﬀerential 293 action (D-brane) 270–3 Born–Infeld form 271, 273 linearized 275–6 action (general) Euclidean 335 fermionic oscillator 343 harmonic oscillator 334, 339–41 in path integral 77, 331, 343 action (point particle) 9–11 Euclidean 82–3 gauge-ﬁxed 129 in curved spacetime 108 Wilson line background 264 action (spacetime) c = 1 moduli 246 closed string tachyon 194 dimensional reduction 232–5, 249, 265 higher-derivative terms 188–9, 195 Hilbert (gravitational) 83, 114, 194 inheritance principle 258 local symmetries 196 massless closed string 114–15 quadratic 275–6 open string 188–9, 196–7, 276 string ﬁeld 311–14, 327–8 tadpole terms 197–8, 225, 309 action (world-sheet) bc 50 βγ 52

Brink–Di Vecchia–Howe–Deser– Zumino 12 Euclidean 82–3 Euler term (χ) 15–16, 30, 83–4, 179, 203 Faddeev–Popov ghost 89, 157–8 gauge-ﬁxing 126–7, 131 Hilbert (gravitational) 15–16 linear dilaton 49 Liouville 324–5 massless scalar 32, 34, 346 Nambu–Goto 10–13, 15, 28, 30 and D-brane action 270 noncritical string 325–6 nonlinear sigma model 108–10, 112 T -duality 280–1 Polyakov 12–16, 28 and noncritical string 325 Euclidean 82–3 ﬂat world-sheet 32 wrapped torus 237 antisymmetric tensor background 249–50 aﬃne Lie algebra 243 annulus, see cylinder anomaly BRST 132–3, 143 from string loops 308–9 conformal same as Weyl current conservation 119, 158–9 ghost number 158–9, 177 gravitational (diﬀ) 95, 349

389

390 anomaly (cont.) Lorentz 23–4, 94 trace same as Weyl Weyl 90–7, 119, 349 from string loops 308–9 in nonlinear sigma model 111–12 in vertex operators 102–7 integrated 93–4 anticommuting c-numbers 341 antiperiodic boundary conditions in traces 212, 336, 344, 345–6 in twisted states 257–9 antisymmetric tensor ﬁeld as axion 29 background 108–15, 120 constant 249–51, 254–5 coupling to D-brane 270–2, 282 winding strings 241–2 dimensional reduction 234–5, 241 ﬁeld equation 111–12 gauge invariance 27, 109, 196 in BRST quantization 137 in light-cone quantization 27–30 in OCQ 125 spacetime action 114 vertex operator 104–7 axion 1, 6, 29 bc CFT 50–1, 75–6, 89 expectation values 176–8, 212–13 mode expansion 61–3 state–operator mapping 65–6 βγ CFT 52, 75 BRST invariance of string amplitudes 160–1, 283, 285, 287–90, 351 of vertex operators 164, 165 BRST quantization 126ﬀ, 350 and string ﬁeld theory 311–15 nilpotence 128, 130–2, 138, 143 point particle 129–31 string 131–43 anomaly 132, 143, 308–9 charge 132 current 131

Index equivalence to light-cone 137–41 equivalence to OCQ 141–3 no-ghost theorem 137–41 transformation 131 background charge 170, 204, 208 background ﬁelds 108–18 linearized, and vertex operators 108–11 Batalin–Vilkovisky (BV) formalism 128 Beltrami diﬀerential 162 Beltrami’s equation 162 Berezin integration 341 Jacobian 345 beta function (Euler) 183 beta function (ﬁeld theory) 91, 111–12, 349 string loop corrections 308–9 world-sheet corrections 112, 195 black holes 6–7, 320 Boltzmann weight 325 Borel summation 316, 352 Born–Infeld action 271, 273 bosonic string theory artifacts of 23, 219, 223–4, 246, 275, 306 reason for studying 8 bosonization 303 boundary state 226, 230 CP T symmetry 23 Calabi–Yau compactiﬁcation 352 Campbell–Baker–Hausdorﬀ formula 202 canceled propagator argument 161, 196, 288, 309 Casimir energy 22 in unitary CFT 73 mnemonic 73 central charge 48, 75–6 and BRST anomaly 132, 143 and compactiﬁcation 113, 117, 137 and density of states 213 and gravitational anomaly 95 and ground state energy 73 and Weyl anomaly 92–6

Index bc 51 βγ 52 free scalar 48 in Virasoro algebra 56 linear dilaton 49 positivity 72, 76 Chan–Paton factor 186–7 Chan–Paton states 30, 184–92 and D-branes 268–70 in cylinder amplitude 223 in M¨ obius strip 228 non-Abelian gauge interactions 188–9 unoriented string 189–92 Chern–Simons term 234–5 chiral gauge symmetry 6 closed state (BRST) 128 closed string 14 BRST quantization 136–7 light-cone quantization 25–9 OCQ 124–5 one-loop amplitudes 216–22, 226–8 poles in cylinder amplitude 225–6 230, 274–6 tree-level amplitudes 192–8 relation to open 195 vertex operators 63–8, 101–7 cocycle 239 coeﬃcient functions (OPE) 37–8 and conformal bootstrap 304 and conformal invariance 47 and matrix elements 200 and three-point amplitudes 284 determine CFT 303 generalized 303 interchange symmetry 40–1 cohomology (BRST) 128–9, 134–43, 144, 350 and amplitudes 283, 285, 289–90 Coleman–Weinberg formula 221, 229–30, 350 collective coordinates (D-brane) 269, 281–2 commutators and anticommutators canonical 17, 20, 26, 130 Jacobi identity 128, 133, 143

391

relation to OPE 54–5 wrong-sign 27–8, 72, 121 commutators and anticommutators (speciﬁc) [αµm , ανn ] 20, 26–7, 58, 62 {bn , cn } 61, 130, 133 [δτ1 , δτ2 ] 129 [GI , GJ ] (constraints) 133 [ jmi , jnj ] 243 [Lm , Ln ] 56–7, 75–6 [Lm , On ] 56 [N g , bm ] 62 [N g , cm ] 62 [pµ , xν ] 17, 20, 26, 58, 62, 130 [Q, A ] 55 [Q1 , Q2 ] 55 {QB , bm } 132 {QB , Lm } 143 {QB , QB } 132–3, 143 [ XL (z), XL (z ) ] 239 compact CFT 117, 137 compactiﬁcation 2, 6, 115–18 abstract CFT 117–18, 137 scale 110, 112–13, 116, 244–5 see also orbifold, orientifold, toroidal complex coordinates 33–4, 153 divergence theorem 33–4 complex manifold 153 conformal bootstrap 304 conformal ﬁeld theory (CFT) 43–76, 349, 351 and compactiﬁcation 115–18 c = 1 261–3, 351 compact 117, 137 coupling to curved metric 95, 97 expectation values and matrix elements 199–201 M¨ obius invariance 198–9, 205 positive genus 169–78, 201–2 zero genus 208–16 free 32–6, 49–52 matter 122, 131 mode expansions 58–63 on Riemann surface 154 periodic scalar 235–40, 242–4, 247–8

392

Index

conformal ﬁeld theory (CFT) (cont.) sewing and cutting 300–5 state–operator isomorphism 63–8 unitary 71–3, 117, 137 see also bc, βγ, free scalar, linear dilaton CFTs conformal gauge 85 and light-cone gauge 83 curvature 93 ghost action 89 OCQ 121 sphere 166 variations around 152 conformal invariance 43–5, and OPE 45–7 relation to diﬀ×Weyl invariance 85–6 conformal Killing equation 151 conformal Killing group (CKG) or vector (CKV) 149–52, 155, 157, 161 speciﬁc Riemann surfaces positive genus 167–9, 176, 178, 192, 197, zero genus 206–8, 212, 216–17 conformal transformation 44–5, 75 and Virasoro algebra 52–8 d ≥ 3 75 of energy-momentum tensor 47–9 of ghost-number current 51 conjugacy classes 260 constraints (in quantization) 28, 130, 133, 143–4, 350 contraction 39–40, 50, 65, 75, 172, 175 self- 103, 171, 338 coordinate transformation large see modular invariance spacetime 28, 109, 196, 285, 310 world-sheet see diﬀ invariance cosmological constant spacetime 222, 306, 350 world-sheet 95, 323 covariant derivative 89, 118 covariant gauge see conformal gauge critical dimension eﬀect of backgrounds 113, 120

from BRST anomaly 132, 143 from cylinder amplitude 230 from Lorentz anomaly 21–4, 27 from modular invariance 230 from OCQ 124, 143 from Veneziano amplitude 204 from Weyl anomaly 94–5 critical point 44 cross-ratio 199 cross-cap 101–2, 207–8, 227–8 state 230 current algebra 243–4 curved spacetime 108–18 cycle A- and B- 147–8, 292, 296, 304 cylinder (annulus) 78, 81 long see handle, degenerating Riemann surface 207 semi-inﬁnite 63–4 string amplitude 222–6, 229–30 calculation of D-brane tension 274–6 closed string poles 225–6, 230 divergence 224–5, 229 unoriented 226 D-brane 7, 31, 104, 268–82, 317, 351 action 270–3 gauge symmetry 268–73 half-D-brane 280 tension 273–7 Dedekind eta function 210, 216 Del Guidice–Di Vecchia–Fubini (DDF) operators 143, 240 density of states 213, 321 determinant functional 182, 338–41, 344–5 in Born–Infeld action 271 of Laplacian 170 variation 12 see also Faddeev–Popov determinant diﬀ (reparameterization) invariance 12–13 anomaly 91, 95, 349 gauge-ﬁxing 85, 129, 325

Index of vertex operators 102–3 relation to conformal invariance 44, 85–6 removes unphysical states 27–8, 117 with boundary 90 dilaton 6 and equivalence principle 115 and string coupling 112–15 background 108–15, 233–4 coupling to D-brane 270–7 open strings 196–8 orientifold plane 280 dimensional reduction 233–4, 241 eﬀect of T -duality on 248 ﬁeld equation 111–12, 246 in BRST quantization 137 in light-cone quantization 27, 29–30 in OCQ 125 linear see linear dilaton CFT moduli space 254 propagator (ﬁeld theory) 276 spacetime action 114–15, 196–8, 225, 275–6 vertex operator 104–7 dimensional reduction 235, 249 Dirichlet boundary condition 31, 266–70, 281 T -dual to Neumann 266–70 discrete gauge symmetry 260–1, 277–8 disk conformal to semi-inﬁnite cylinder 63–8 expectation values 174–5, 178 Riemann surface 168 string amplitudes 178–89, 196–8, 204–5 with D-branes 270, 274, 281–2 tadpole 197–8, 223–6 cancellation with projective plane 229 divergences analytic continuation past 181, 287 cancellation for SO(8192) 228–9 from boundary of moduli space 196, 296, 305–6

393

IR 181, 223–5, 306–10 UV 2–7, 218–20 absence in string theory 4–5, 79, 219, 223–5, 309–10 zero mode 173 doubling trick 57, 63, 66, 178 dual resonance model 348 duality see T -duality, world-sheet duality. eﬀective ﬁeld theory 110, 114, 117, 188–9, 196–7, 270–3, 309 Einstein metric 115, 194, 275 Einstein’s equation 112–13, 222 electroweak scale 222 energy-momentum tensor 13, 91 and derivative with respect to moduli 161, 210–11 as constraint 121 boundary condition 56–7 choice of 49, 95, 303 conformal transformation 47–9 trace anomaly 92–7 traceless and holomorphic 43–4 vanishes for Chan–Paton states 185 enhanced gauge symmetry 242–4, 248, 254 epsilon prescription 182, 287–8, 305–6 equivalence classes BRST 128 gauge 84, 87 OCQ 123 equivalence principle 115 Euclidean adjoint 202–3, 336 Euclidean continuation 82–3, 170, 210, 335 Euler number 96, 101, 118, 152 in string action 83–4, 112, 179, 203 exact state (BRST) 128 exceptional Lie algebras 192 expectation values 34, 36 calculations 169–78, 201–2, 208–13 exponential operator 40, 46 exterior derivative 128

394 Faddeev–Popov determinant 86–90, 154–8 on disk 176 on sphere 176 on torus 212–13, 216–17 shortcuts 179, 218 sign 156, 177, 345 Faddeev–Popov ghosts 51–2, 88–90, 126 boundary condition 90 central charge 94 vacuum 61, 141 Faddeev–Popov procedure, general 126 Feynman diagrams calculation of D-brane interaction 275–6 calculation of Weyl anomaly 119 decomposition of string amplitudes 297–306 vs string graphs 3–5, 101 ﬁducial metric 85 ﬁeld strength, three-index 109, 112, 234–5 ﬁrst-quantization 310 Fischler–Susskind mechanism 308–9 ﬁxed point orbifold 256–8 UV 4 ﬂat direction 246, 262–3, 272–3 free energy 321–2, 328 free parameters, absence of 6, 83–4, 112–13 free scalar CFT 32–41 expectation values 169–76, 208–11, 229 left-, right-moving 238 mode expansion 58–61 state–operator mapping 63–5 Fuchsian group 292–3 functional integral see path integral fundamental region 148–50 in torus amplitude 216, 219–20 gauge boson 1, 4 open string 28, 123–4, 135–6

Index amplitudes 187–9, 205 in BRST quantization 135–6 in light-cone quantization 23 in OCQ 123–4 vertex operator 107 closed string 241–3 amplitudes 242 gauge coupling 115, 234, 242–5, 276 gauge-ﬁxing dependence on gauge choice 90–1, 121–2, 126–8 diﬀ×Weyl 17–18, 25, 84–6, 131, 154–5 general 126–8 see also speciﬁc gauges gauge interaction 2, 28, 78, 184–92 gauge transformation 107 Gaussian-averaged gauges 127 general relativity 1–2, 5, 28, 108 oﬀ-shell amplitudes 104 genus 100–1, 150 large 315–17 geodesic curvature 30, 82, 96 geodesic distance 102 geodesic equation 108 ghost number 51, 62, 127 anomaly 119, 158–9, 177 of string ﬁeld 313–14 states vs operators 65–6 ghosts see Faddeev–Popov ghosts global symmetry 189 Goldstone boson 233 graded Lie algebra 143 grand uniﬁcation 2, 5 Grassmann variables 35, 88, 156–8, 341–5 gravitational coupling 114 relation to gauge coupling 234, 244–5 string coupling 194, 205 graviton coupling to D-brane 270–7 open strings 196–8 orientifold plane 280 in BRST quantization 137

Index in light-cone quantization 27, 29–30 in OCQ 125 propagator (ﬁeld theory) 276 vertex operator 104–7 gravity 1–7, 78 Green’s function 169–71, 174–6, 208, 229–30, 275 Hagedorn temperature 321–2, 352 Hamiltonian and path integral 329–32, 335–6, 343 eﬀect of backgrounds 250, 264 in partition function 209, 212–13 point particle 17, 130, 218 string 19–21, 25–6 relation to L0 54, 73, 125 handle 83, 100–1, 290–2 degenerating 97–8, 101, 224–6, 229, 295–6, 305–9 hard scattering 183–4, 194, 281–2, 317–19, 352 heterotic string 348 hidden sector 6 Higgs ﬁeld 1, 222, 245–6 high-energy scattering see hard scattering, Regge scattering high temperature 320–2 highest weight state 70–2, 122, 125, 143 and tensor (primary) operators 71 holomorphic current 42 ﬁeld 34, 56, 72 quadratic diﬀerential 152 transition functions 153–4 vector ﬁeld 152 hyperelliptic surface 294 infrared (IR) divergences 181, 223–5, 306–10 inheritance principle 258, 261 inner product bilinear (Zamolodchikov) 199–204 in sewing 301–2 relation to Hermitean 202–4 Fermi oscillator 343 Hermitean 134–5

395

positive 71–2, 116–17, 138 traceless symmetric tensor 118 instantons 316 interaction irrelevant 3 string 77–82, 104, 178ﬀ Jacobi identity 128, 133, 143 Jacobi’s abstruse identity 215 Jacobian, Berezin integral 345 Kac–Moody algebra 243 Kaluza–Klein gauge ﬁeld 2, 232–4, 241–3, 249, 257, 351 Kaluza–Klein states 232, 245 Klein bottle Riemann surface 207 string amplitude 226–9, 230, 278, 280 large order behavior 315–17 lattice even self-dual 252–3, 281 momentum 251–3, 281 Laurent expansion see mode expansion length, minimum 248, 318 level current algebra 243–4 oscillator 23, 26–7 level-matching 27, 258 light-cone coordinates 16 light-cone gauge 17 oﬀ-shell 104 light-cone quantization point particle 17 string 17–30, 349 equivalence to BRST and OCQ 137–43 linear dilaton theory 49–50, 61, 75, 204, 213, 349 as background 113, 120, 323–4, 328 Liouville ﬁeld theory 325 loop expansion 115 Lorentz invariance 12, 106–7, 116–17, 290 in light-cone gauge 20–4, 90, 94, 349

396

Index

Mandelstam variables 180 manifold complex 153 diﬀerentiable 153 Riemannian 153 mass-shell condition from vertex operators 101, 106 in BRST 131, 134, 137–8 in light-cone quantization 17, 22, 26 matrix elements, relation to path integral 199–201, 332–4 matrix models 326–7, 352 measure on moduli space 154–65 BRST invariance 160–1 derivation 154–7 gauge invariance 159–60 in terms of determinants 157–8 transition functions 161–4 see also Faddeev–Popov metric (spacetime) 9, 108–18 dimensional reduction 231–3 Einstein 115, 194, 275 internal 249ﬀ light-cone coordinates 16 sigma model 115 string 115 metric (world-line) 10 gauge-ﬁxing 129, 145–6 metric (world-sheet) constant curvature 292 ﬂat 32–3 gauge-ﬁxing 17, 84–6, 146–52 induced 11, 15 minimal area 300, 351 noncritical string 94–5, 325–6 Polyakov 12–13ﬀ mnemonics, zero-point 73–4, 125 M¨ obius group 56, 167 M¨ obius strip Riemann surface 208 string amplitude 228–9, 280 M¨ obius transformation 178–80, 192, 198–9 mode (or Laurent) expansion bc CFT 61

free scalar (X µ ) 26, 58 periodic 235–8 twisted 257 holomorphic operator 56 open strings 20, 57, 62–3 Dirichlet 267 Tab 53, 70 model, solvable 323–7 modular group 148–9, 151 and theta functions 214–16 spacetime 253–5 surfaces without 207–8, 223, 278 used to analyze asymptotics 213, 224, 227–8 modular invariance 210, 217–18, 229–30 constraints from 213, 304, 350 in orbifold compactiﬁcation 257, 259 in toroidal compactiﬁcation 237, 251–3 in twisting construction 260 needed for consistency 220 one-point function 304 moduli (spacetime) 233–4, 249, 265 moduli space 253–5 branches 262 c = 1 261–3, 351 metric on 254 moduli (world-sheet) encoded in transition functions 152–4, 161–4 for metric 147–52 moduli space 148–52, 290–4 boundary of 161, 295–6, 300; CFT asymptotics 302–3; divergences from 196, 224–6, 300, 305–9 graphical decomposition 297–300, 351 in open string ﬁeld theory 313–4, 316–7 measure on 154–65, 350 total derivative on 161, 196, 220, 288

Index vertex operator positions 150–1, 163–4 Nambu–Goto action 10–13, 15, 28, 30 Narain compactiﬁcation 251–5, 351 naturalness problem 2 Neumann boundary condition 14, 31, 62, 174 T -dual to Dirichlet 265–70 nilpotence see BRST no-ghost theorem 137–43, 350 Noether’s theorem 41–3, 74–5 anomalous 159 as local property 86 non-Abelian gauge theory 188–92, 243 from string ﬁeld theory 313 on D-brane 272–3 noncommutative geometry 272 noncritical string theory 325–7 nonlinear sigma model 108–10, 120, 349 nonperturbative eﬀects 234, 248, 269, 310, 315–7 exp(−1/gc ) 317, 327 normal ordering boundary 75, 175 conformal 36–40, 50, 60 constants 22–3, 27, 76 direct calculation 60 from state–operator mapping 72 from Virasoro algebra 59, 61 mnemonics 73, 125 creation–annihilation 60 diﬀ-invariant 102–3 relation between diﬀerent forms 59–60, 76 null state BRST 128, 136, 311 decoupling 119, 161, 196, 220, 285–95 OCQ 122–5, 143 O(k, k, R) 252–3 O(k, k, Z) 253 oﬀ-shell amplitudes 97, 103–4 old covariant quantization (OCQ) 121–5

397

equivalence to BRST and light-cone 141–3 vertex operators 125, 143, 160–1, 164, 165 one-loop amplitudes 150, 206–30 ﬁniteness 220 vs ﬁeld theory 218–20 open string theory 14 BRST quantization 131–6 Dirichlet 266–70 light-cone quantization 17–25 must include closed strings 81–2, 225 OCQ 122–5 one-loop amplitudes 222–6, 228–9 tree–level amplitudes 178–92 relation to closed 195 vertex operators 66, 107 operator calculations 201–2 operator equation 35–6, 333–4 operator product expansion (OPE) 36–41, 74–6 and conformal invariance 45–9 and Ward identity 42–3 associativity 70, 201, 304 convergence 37–8 derivation CFT 68–70 free ﬁeld theory 37–8 generalized 302–3 interchange symmetry 40–1 Laurent expansion 70–1 nonzero separation 48 relation to commutator 54–7 z-, ¯z -dependence 47 operator product expansions (speciﬁc) Ai Aj 47 b b, b c, c c 50 β β, β γ, γ γ 52 DDF operators 240 ∂X µ ∂Xµ 211 : eik·X : : eik ·X : 40, 251 : F : : G : 39 jB b, jB c 132 jB Om 132 jB jB 132

398

Index

operator product expansions (speciﬁc) (cont.) jB V 165 j i j j 243 j µ : eik·X : 43 Oi Oj 200 T A 45–6 T eik·X 46 T j 51 T jB 133 T O 46 T T 47–8 ˜ 47–8 TT V V 284, 289, 327 XL XL , XL XR , XR XR 238 X µ X ν 38 orbifold 256–63, 351 ﬁxed points 256–8 inheritance principle 258, 261 modular invariance 259–60 twisted states 31, 256–61 vertex operators 258, 351 twisting construction 259–63 oriented string theory 29 see complement of unoriented string theory orientifold 277–80 ﬁxed planes 278–80 local physics oriented 278 outline of both volumes 8–9 P SL(2, C) 168, 192, 197–9, 205 P SL(2, R) 168, 178, 199, 205, 255, 292–3 P SL(2, Z) 148, 255 parity spacetime 6, 255, 277 world-sheet 29, 200, 255, 345 action on Chan–Paton factors 189–92 gauging 277 in one-loop amplitudes 207–8, 226–8 partition function 209 bc 212 general CFT 213

orbifold 258–61 scalar CFT 209–11 thermal 320–2, 328 toroidal compactiﬁcation 237, 250–2, 281 path (functional) integrals 32–6 and expectation values 34 canonical commutators from 54–5, 58–61 Euclidean rotation 82–3, 170, 210, 335 evaluation by integrating Weyl anomaly 93–4 by operator methods 201–2, 209–10, 212 direct 157–9, 169–72, 176–7, 337–41, 344–5 using holomorphicity 172–4, 177, 210–11 existence 332 fermionic 341–5 for traces 335–6, 344 Hamiltonian 331 introduction 329–46 Lagrangian 331 of total derivative 34, 333, 341 operator equations in 35, 333 Polyakov 82–4 gauge-ﬁxing 84–90, 154–9 in curved spacetime 108–12 Weyl anomaly in 90–7 relation to operator expressions via cutting 55, 199–201, 300–5, 332–4 representation of string amplitudes 77–82, 97–101 twisted 260, 336 used to derive state–operator mapping 66–8 Pauli–Villars regulator 91, 340 period matrix 293 periodic boundary condition 14, 31, 52 perturbation theory, string 5, 101, 283–310 physical state BRST 127–9

Index b0 condition 131, 134, 136, 289–90 vertex operators 164 OCQ 121–3 L0 condition 125 vertex operators 125, 143, 160–1, 164 Planck length 3, 116 Planck mass 3, 245 plumbing ﬁxture 294–305 Poincar´e invariance 10, 12–16, 91, 185 and tadpoles 225 four-dimensional 16 of boundary conditions 14–15 point particle action 9–11 and Wilson line 264 BRST quantization 129–31 in curved spacetime 108 light-cone quantization 17 path integral 77, 82–3, 164–5 vacuum loop 145–6, 164, 218, 220–2 Poisson resummation formula 237, 250, 252 Polyakov action see action, Polyakov Polyakov path integral see path integral, Polyakov potential energy c = 1 moduli 246 from dilaton tadpoles 197–8, 225, 309 from quartic Yang–Mills interaction 265, 272 primary ﬁeld see tensor operator projective plane expectation values 176, 178 Riemann surface 168–9 string amplitudes 196–8 tadpole 197–8, 227–8 cancellation with disk 229 propagator dilaton 275–6 graviton 275–6 photon 290 point particle 115, 164–5 pole 225–6, 307–8

399

string 297, 305–6, 313–14 proton decay 1 puncture 296–300 QCD string 25 quantization canonical 17, 19–20, 26, 130, 249–50, 164 path integral 32–6, 77–82, 329–46 see also BRST, light-cone, OCQ, path integrals quantum ﬁeld theory 1–7, 104, 113, 310, 331–2 UV divergence 218–20 world–sheet 13–15, 32ﬀ, 108–10, 118 quantum gravity 2–7, 25 quantum mechanics 1, 116, 329–46 Regge behavior 183–4, 193, 320, 352 Regge slope 11, 24, 112 renormalization 38, 102–3, 189, 332, 340 renormalization group equation 4, 91, 111 Ricci curvature spacetime 111–12, 114 dimensional reduction 232–3 world-sheet 15, 85, 92–3 Riemann–Roch theorem 152, 158–9, 293 Riemann surface 100–1, 152–4, 161–4, 351 sewing 294–300, 351 speciﬁc 166–9, 206–8, 290–4 Riemann tensor 85 SL(2, C) (algebra) 71 invariant state 71, 203 SL(2, R) (algebra) 56, 71 SO(D − 2) 23–4, 27, 30 SO(D − 1) 23–4, 27, 30–1 SO(n) 30, 191–2, 228 on D-brane 279–80 SO(8192) 228–9, 280 Sp(k) 30, 191–2, 228, 279 SU(n) 189

400

Index

SU(2) × SU(2) radius 242–8, 261–3 on D-brane 264–5, 269–72, 279 SU(3) × SU(2) × U(1) model 1 see also Standard Model S-matrix 97–101, 103–4 calculations 178–205 expression for 99, 155–7, 164 unitarity 181–2, 186–7, 193–4, 196, 285–90, 306 vs Green’s function 104 s, t, u (Mandelstam variables) 180 saturating geodesic 300 scale transformation 44–8, 90–1 Schottky group 290–1, 351 Schwarzian 48 Schwinger–Dyson equation 334 self-contractions 103, 171, 338 self-dual radius 242–8, 261–3 sewing 332, 351 circles 298–9 construction of general surface from three-punctured spheres 296–7 of CFTs 300–5 and conformal bootstrap 303–5 of Riemann surfaces 294–300 parameter 294–5 used to analyze string amplitudes 305–7 Siegel gauge 313 sigma model metric 115 soliton 104, 235 sphere expectation values 174–8, 198–204 Riemann surface 166–8 string amplitudes 192–6, 204–5, 283–90 thrice-punctured 296–7, 303 cancellation with projective plane 229 spin 23–4, 184, 204 in CFT 46, 213 in Coleman–Weinberg formula 221 spurious state 122 Standard Model 1–7, 25, 112, 310, 348–9

state–operator isomorphism 63–73, 101–2 explicit form 65 path integral derivation 66–8 vertex operators 101–2, 164, 199–201 Stirling’s approximation 184, 315 string coupling 83–4 and dilaton 112–13 as overall factor in action 115 in vertex operators 101–2, 107, 179 position-dependent 323–4 relation between open and closed 83–4, 276 relation to gauge coupling 187, 234 relation to gravitational coupling 194, 205 string ﬁeld theory 310–17, 352 string length scale 25, 110, 244, 248 string metric 115 string tension 11, 30 subtraction 39, 47, 60 supersymmetry 2, 6–7, 222, 283, 315, 348 supersymmetry breaking 222, 233–4 T -duality 247–8, 280–1 and D-branes 268–73 as gauge symmetry 248 O(k, k, Z) 253–5 open string theory 265–8 reverses winding and momentum 247, 268 unoriented string theory 277–80 tachyon amplitudes 178–84, 187, 192–5, 204–5, 217, 229–30 artifact of bosonic string theory 23, 219, 223–4, 246, 275, 306 in BRST quantization 135 in light-cone quantization 23, 27, 29–30 in linear dilaton background 323–4, 328 in OCQ 123

Index present in all bosonic string theories 117 vertex operator 101, 107 tadpole 197, 309 cancellation for SO(8192) 229 disk 197, 223–6, 228–9 projective plane 197, 227–9 torus 307 target space 109 Teichm¨ uller parameters 148 temperature 320–2, 328 tensor operator (primary ﬁeld) 46–7, 76 expectation values 199 maps to highest weight state 71 OPEs 47, 200, 303 tetrad spacetime 250–1 world-line 10, 129, 145–6 world-sheet 83 theta functions 214–16, 229, 350 in expectation values 208–9, 211, 228, 259 toroidal compactiﬁcation 231–55, 236–80, 280–1, 351 abstract (Narain) description 251–4 in CFT 235–40 in ﬁeld theory 231–5 in string theory 241–8 of several dimensions 249–55 of two dimensions 255 open string 263–8 torus and partition function 209 expectation values 208–13 one-point function 304 in twisted CFT 260 Riemann surface 146–50, 154, 206–7 string amplitude 216–22, 229–30, 237, 250 ﬁniteness 218–20 trace of energy-momentum tensor 13, 43, 92–7, 111 partition function 209, 335–6, 344, 345–6

401

transition functions 153–4, 161–4 unoriented 277 transverse Hilbert space 138 oscillators 20–1, 26, 125, 217–18 tree-level amplitudes 178–98, 283–90, 350–1 inheritance principle 258, 261 open–closed relation 195, 350 poles 181–2, 193–4, 196, 287–8 twisted ﬁelds 31, 257, 259, 336, 345–6 twisted state 31, 257–61 twisting (orbifold) construction 259–63 U(n) 30, 185, 188–9, 230 ultraviolet (UV) divergences see under divergences unit gauge 85 unitarity see under S-matrix unitary CFT 71–3, 117, 137, 213 unoriented string theory 29–30, 80–1 Chan–Paton states 189–92 one-loop amplitudes 226–9 orientation-reversal as gauge symmetry 149, 153, 277–9 Riemann surfaces 80–1, 100–1 T -duality 277–80, 351 vacuum amplitude ﬁeld theory 218 physics of 220–2 string 198, 216–30, 274–6 vacuum degeneracy 7, 118, 233–4, 261, 348 Veneziano amplitude 178–84, 204, 348 vertex operators 63–8, 74, 97–102, 349 general construction 63–6 must be on mass-shell 101–3, 113 normalization 101–2, 107, 187, 194, 276, 284 OCQ 125, 143, 160–1, 164, 165 Polyakov approach 102–7, 119 positions 99–100 coincident limit 195–6, 306–9 in speciﬁc amplitudes 178–80, 192–3, 204, 216–17

402 vertex operators (cont.) positions (cont.) relation to moduli and CKVs 149–52, 154–7, 163–4 speciﬁc 101–2, 107 twisted state 258, 351 winding state 238–9, 243–5, 251 Virasoro algebra 52–7, 75–6 Virasoro generators as constraints 121–2, 133 centered on operators 70 L−1 · as derivative 71 L0 · as weight 71 Hermiticity 71 oscillator expressions 59, 61–2 winding state 238 Virasoro–Shapiro amplitude 192–5, 204–5 Ward identity 41–3, 55 as local property 86 for conformal symmetry 45–6 weight (conformal) 46–7, 71–3 of vertex operators 101, 125, 143 positivity 72 Wess–Zumino consistency condition 96 Weyl anomaly see under anomaly Weyl transformation 13, 15–16, 30 as redundancy 13, 84, 117, 326 gauge-ﬁxing of 17–18, 85 ghosts trivial 88, 131

Index of curvature 15, 85, 114 of nonlinear sigma model 110–12 of vertex operators 102–7 relation to conformal transformation 86 spacetime 114, 233, 254 Wick’s theorem 60 Wilson line 263–8, 279–80 breaks gauge symmetry 265, 270 winding state 31, 235–9 Dirichlet open string 266–7 massless as R → 0 247 vertex operators 238–9, 243–5, 251 world-sheet 13–14 world-sheet duality 287, 332 cylinder 226, 274 wrapped D-brane 273 world-sheet 237, 250 Yukawa couplings 1, 79 Zamolodchikov inner product 200 zero modes ghost 157–9, 161, 176, 212 X µ 169–70, 173 zero-norm states see null states zero-point energy 59–61, 72 heuristic 21–2, 26 mnemonics 73, 125 zero-slope limit 188

String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal ﬁeld theory. The ﬁrst four chapters introduce the central ideas of string theory, the tools of conformal ﬁeld theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactiﬁcation and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the ﬁniteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersymmetric string theories and goes on to a broad presentation of the important advances of recent years. The ﬁrst three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal ﬁeld theory. The ﬁnal four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries. An appendix summarizes the necessary background on fermions and supersymmetry. Both volumes contain an annotated reference section, emphasizing references that will be useful to the student, as well as a detailed glossary of important terms and concepts. Many exercises are included which are intended to reinforce the main points of the text and to bring in additional ideas. An essential text and reference for graduate students and researchers in theoretical physics, particle physics, and relativity with an interest in modern superstring theory. Joseph Polchinski received his Ph.D. from the University of California at Berkeley in 1980. After postdoctoral fellowships at the Stanford Linear Accelerator Center and Harvard, he joined the faculty at the University of Texas at Austin in 1984, moving to his present position of Professor of Physics at the University of California at Santa Barbara, and Permanent Member of the Institute for Theoretical Physics, in 1992. Professor Polchinski is not only a clear and pedagogical expositor, but is also a leading string theorist. His discovery of the importance of D-branes in 1995 is one of the most important recent contributions in this ﬁeld, and he has also made signiﬁcant contributions to many areas of quantum ﬁeld theory and to supersymmetric models of particle physics.

From reviews of the hardback editions: Volume 1 ‘. . . This is an impressive book. It is notable for its consistent line of development and the clarity and insight with which topics are treated . . . It is hard to think of a better text in an advanced graduate area, and it is rare to have one written by a master of the subject. It is worth pointing out that the book also contains a collection of useful problems, a glossary, and an unusually complete index.’ Physics Today ‘. . . the most comprehensive text addressing the discoveries of the superstring revolutions of the early to mid 1990s, which mark the beginnings of “modern” string theory.’ Donald Marolf, University of California, Santa Barbara, American Journal of Physics ‘Physicists believe that the best hope for a fundamental theory of nature – including unification of quantum mechanics with general relativity and elementary particle theory – lies in string theory. This elegant mathematical physics subject is expounded by Joseph Polchinski in two volumes from Cambridge University Press . . . Written for advanced students and researchers, this set provides thorough and up-to-date knowledge.’ American Scientist ‘We would like to stress the pedagogical value of the present book. The approach taken is modern and pleasantly systematic, and it covers a broad class of results in a unified language. A set of exercises at the end of each chapter complements the discussion in the main text. On the other hand, the introduction of techniques and concepts essential in the context of superstrings makes it a useful reference for researchers in the field.’ Mathematical Reviews ‘It amply fulfils the need to inspire future string theorists on their long slog and is destined to become a classic. It is a truly exciting enterprise and one hugely served by this magnificent book.’ David Bailin, The Times Higher Education Supplement Volume 2 ‘In summary, these volumes will provide . . . the standard text and reference for students and researchers in particle physics and relativity interested in the possible ramifications of modern superstring theory.’ Allen C. Hirshfeld, General Relativity and Gravitation ‘Polchinski is a major contributor to the exciting developments that have revolutionised our understanding of string theory during the past four years; he is also an exemplary teacher, as Steven Weinberg attests in his foreword. He has produced an outstanding two-volume text, with numerous exercises accompanying each chapter. It is destined to become a classic . . . magnificent.’ David Bailin, The Times Higher Education Supplement ‘The present volume succeeds in giving a detailed yet comprehensive account of our current knowledge of superstring dynamics. The topics covered range from the basic construction of the theories to the most recent discoveries on their non-perturbative behaviour. The discussion is remarkably self-contained (the volume even contains a useful appendix on spinors and supersymmetry in several dimensions), and thus may serve as an introduction to the subject, and as an excellent reference for researchers in the field.’ Mathematical Reviews

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STRING THEORY VOLUME I An Introduction to the Bosonic String JOSEPH POLCHINSKI Institute for Theoretical Physics University of California at Santa Barbara

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Contents

Foreword

xiii

Preface

xv

Notation

xviii

1 A ﬁrst look at strings 1.1 Why strings? 1.2 Action principles 1.3 The open string spectrum 1.4 Closed and unoriented strings Exercises

1 1 9 16 25 30

2 Conformal ﬁeld theory 2.1 Massless scalars in two dimensions 2.2 The operator product expansion 2.3 Ward identities and Noether’s theorem 2.4 Conformal invariance 2.5 Free CFTs 2.6 The Virasoro algebra 2.7 Mode expansions 2.8 Vertex operators 2.9 More on states and operators Exercises

32 32 36 41 43 49 52 58 63 68 74

3 3.1 3.2 3.3 3.4 3.5 3.6

77 77 82 84 90 97 101

The Polyakov path integral Sums over world-sheets The Polyakov path integral Gauge ﬁxing The Weyl anomaly Scattering amplitudes Vertex operators

ix

x

Contents

3.7 Strings in curved spacetime Exercises

108 118

4 The string spectrum 4.1 Old covariant quantization 4.2 BRST quantization 4.3 BRST quantization of the string 4.4 The no-ghost theorem Exercises

121 121 126 131 137 143

5 The string S-matrix 5.1 The circle and the torus 5.2 Moduli and Riemann surfaces 5.3 The measure for moduli 5.4 More about the measure Exercises

145 145 150 154 159 164

6 Tree-level amplitudes 6.1 Riemann surfaces 6.2 Scalar expectation values 6.3 The bc CFT 6.4 The Veneziano amplitude 6.5 Chan–Paton factors and gauge interactions 6.6 Closed string tree amplitudes 6.7 General results Exercises

166 166 169 176 178 184 192 198 204

7 One-loop amplitudes 7.1 Riemann surfaces 7.2 CFT on the torus 7.3 The torus amplitude 7.4 Open and unoriented one-loop graphs Exercises

206 206 208 216 222 229

8 Toroidal compactiﬁcation and T -duality 8.1 Toroidal compactiﬁcation in ﬁeld theory 8.2 Toroidal compactiﬁcation in CFT 8.3 Closed strings and T -duality 8.4 Compactiﬁcation of several dimensions 8.5 Orbifolds 8.6 Open strings 8.7 D-branes 8.8 T -duality of unoriented theories Exercises

231 231 235 241 249 256 263 268 277 280

Contents

xi

9 Higher order amplitudes 9.1 General tree-level amplitudes 9.2 Higher genus Riemann surfaces 9.3 Sewing and cutting world-sheets 9.4 Sewing and cutting CFTs 9.5 General amplitudes 9.6 String ﬁeld theory 9.7 Large order behavior 9.8 High energy and high temperature 9.9 Low dimensions and noncritical strings Exercises

283 283 290 294 300 305 310 315 317 322 327

Appendix A: A short course on path integrals A.1 Bosonic ﬁelds A.2 Fermionic ﬁelds Exercises

329 329 341 345

References

347

Glossary

359

Index

389

Outline of volume II

10

Type I and type II superstrings

11

The heterotic string

12

Superstring interactions

13

D-branes

14

Strings at strong coupling

15

Advanced CFT

16

Orbifolds

17

Calabi–Yau compactiﬁcation

18

Physics in four dimensions

19

Advanced topics

Appendix B: Spinors and SUSY in various dimensions

xii

Foreword

From the beginning it was clear that, despite its successes, the Standard Model of elementary particles would have to be embedded in a broader theory that would incorporate gravitation as well as the strong and electroweak interactions. There is at present only one plausible candidate for such a theory: it is the theory of strings, which started in the 1960s as a not-very-successful model of hadrons, and only later emerged as a possible theory of all forces. There is no one better equipped to introduce the reader to string theory than Joseph Polchinski. This is in part because he has played a signiﬁcant role in the development of this theory. To mention just one recent example: he discovered the possibility of a new sort of extended object, the ‘Dirichlet brane,’ which has been an essential ingredient in the exciting progress of the last few years in uncovering the relation between what had been thought to be diﬀerent string theories. Of equal importance, Polchinski has a rare talent for seeing what is of physical signiﬁcance in a complicated mathematical formalism, and explaining it to others. In looking over the proofs of this book, I was reminded of the many times while Polchinski was a member of the Theory Group of the University of Texas at Austin, when I had the beneﬁt of his patient, clear explanations of points that had puzzled me in string theory. I recommend this book to any physicist who wants to master this exciting subject. Steven Weinberg Series Editor Cambridge Monographs on Mathematical Physics 1998

xiii

Preface

When I ﬁrst decided to write a book on string theory, more than ten years ago, my memories of my student years were much more vivid than they are today. Still, I remember that one of the greatest pleasures was ﬁnding a text that made a diﬃcult subject accessible, and I hoped to provide the same for string theory. Thus, my ﬁrst purpose was to give a coherent introduction to string theory, based on the Polyakov path integral and conformal ﬁeld theory. No previous knowledge of string theory is assumed. I do assume that the reader is familiar with the central ideas of general relativity, such as metrics and curvature, and with the ideas of quantum ﬁeld theory through nonAbelian gauge symmetry. Originally a full course of quantum ﬁeld theory was assumed as a prerequisite, but it became clear that many students were eager to learn string theory as soon as possible, and that others had taken courses on quantum ﬁeld theory that did not emphasize the tools needed for string theory. I have therefore tried to give a self-contained introduction to those tools. A second purpose was to show how some of the simplest fourdimensional string theories connect with previous ideas for unifying the Standard Model, and to collect general results on the physics of fourdimensional string theories as derived from world-sheet and spacetime symmetries. New developments have led to a third goal, which is to introduce the recent discoveries concerning string duality, M-theory, D-branes, and black hole entropy. In writing a text such as this, there is a conﬂict between the need to be complete and the desire to get to the most interesting recent results as quickly as possible. I have tried to serve both ends. On the side of completeness, for example, the various path integrals in chapter 6 are calculated by three diﬀerent methods, and the critical dimension of the bosonic string is calculated in seven diﬀerent ways in the text and exercises. xv

xvi

Preface

On the side of eﬃciency, some shorter paths through these two volumes are suggested below. A particular issue is string perturbation theory. This machinery is necessarily a central subject of volume one, but it is somewhat secondary to the recent nonperturbative developments: the free string spectrum plus the spacetime symmetries are more crucial there. Fortunately, from string perturbation theory there is a natural route to the recent discoveries, by way of T -duality and D-branes. One possible course consists of chapters 1–3, section 4.1, chapters 5–8 (omitting sections 5.4 and 6.7), chapter 10, sections 11.1, 11.2, 11.6, 12.1, and 12.2, and chapters 13 and 14. This sequence, which I believe can be covered in two quarters, takes one from an introduction to string theory through string duality, M theory, and the simplest black hole entropy calculations. An additional shortcut is suggested at the end of section 5.1. Readers interested in T -duality and related stringy phenomena can proceed directly from section 4.1 to chapter 8. The introduction to Chan– Paton factors at the beginning of section 6.5 is needed to follow the discussion of the open string, and the one-loop vacuum amplitude, obtained in chapter 7, is needed to follow the calculation of the D-brane tension. Readers interested in supersymmetric strings can read much of chapters 10 and 11 after section 4.1. Again the introduction to Chan–Paton factors is needed to follow the open string discussion, and the one-loop vacuum amplitude is needed to follow the consistency conditions in sections 10.7, 10.8, and 11.2. Readers interested in conformal ﬁeld theory might read chapter 2, sections 6.1, 6.2, 6.7, 7.1, 7.2, 8.2, 8.3 (concentrating on the CFT aspects), 8.5, 10.1–10.4, 11.4, and 11.5, and chapter 15. Readers interested in four-dimensional string theories can follow most of chapters 16–19 after chapters 8, 10, and 11. In a subject as active as string theory — by one estimate the literature approaches 10 000 papers — there will necessarily be important subjects that are treated only brieﬂy, and others that are not treated at all. Some of these are represented by review articles in the lists of references at the end of each volume. The most important omission is probably a more complete treatment of compactiﬁcation on curved manifolds. Because the geometric methods of this subject are somewhat orthogonal to the quantum ﬁeld theory methods that are emphasized here, I have included only a summary of the most important results in chapters 17 and 19. Volume two of Green, Schwarz, and Witten (1987) includes a more extensive introduction, but this is a subject that has continued to grow in importance and clearly deserves an introductory book of its own. This work grew out of a course taught at the University of Texas

Preface

xvii

at Austin in 1987–8. The original plan was to spend a year turning the lecture notes into a book, but a desire to make the presentation clearer and more complete, and the distraction of research, got in the way. An early prospectus projected the completion date as June 1989 ± one month, oﬀ by 100 standard deviations. For eight years the expected date of completion remained approximately one year in the future, while one volume grew into two. Happily, ﬁnally, one of those deadlines didn’t slip. I have also used portions of this work in a course at the University of California at Santa Barbara, and at the 1994 Les Houches, 1995 Trieste, and 1996 TASI schools. Portions have been used for courses by Nathan Seiberg and Michael Douglas (Rutgers), Steven Weinberg (Texas), Andrew ˜ Paulo) Strominger and Juan Maldacena (Harvard), Nathan Berkovits (Sao and Martin Einhorn (Michigan). I would like to thank those colleagues and their students for very useful feedback. I would also like to thank Steven Weinberg for his advice and encouragement at the beginning of this project, Shyamoli Chaudhuri for a thorough reading of the entire manuscript, and to acknowledge the support of the Departments of Physics at UT Austin and UC Santa Barbara, the Institute for Theoretical Physics at UC Santa Barbara, and the National Science Foundation. During the extended writing of this book, dozens of colleagues have helped to clarify my understanding of the subjects covered, and dozens of students have suggested corrections and other improvements. I began to try to list the members of each group and found that it was impossible. Rather than present a lengthy but incomplete list here, I will keep an updated list at the erratum website http://www.itp.ucsb.edu/˜joep/bigbook.html. In addition, I would like to thank collectively all who have contributed to the development of string theory; volume two in particular seems to me to be largely a collection of beautiful results derived by many physicists. String theory (and the entire base of physics upon which it has been built) is one of mankind’s great achievements, and it has been my privilege to try to capture its current state. Finally, to complete a project of this magnitude has meant many sacriﬁces, and these have been shared by my family. I would like to thank Dorothy, Steven, and Daniel for their understanding, patience, and support. Joseph Polchinski Santa Barbara, California 1998

Notation

This book uses the +++ conventions of Misner, Thorne, and Wheeler (1973). In particular, the signature of the metric is (− + + . . . +). The constants ¯h and c are set to 1, but the Regge slope α is kept explicit. A bar ¯ is used to denote the conjugates of world-sheet coordinates and moduli (such as z, τ, and q), but a star ∗ is used for longer expressions. A bar on a spacetime fermion ﬁeld is the Dirac adjoint (this appears only in volume two), and a bar on a world-sheet operator is the Euclidean adjoint (deﬁned in section 6.7). For the degrees of freedom on the string, the following terms are treated as synonymous: holomorphic = left-moving, antiholomorphic = right-moving, as explained in section 2.1. Our convention is that the supersymmetric side of the heterotic string is right-moving. Antiholomorphic operators are designated by tildes ˜; as explained in section 2.3, these are not the adjoints of holomorphic operators. Note also the following conventions: d2 z ≡ 2dxdy ,

1 δ 2 (z, ¯z ) ≡ δ(x)δ(y) , 2

where z = x + iy is any complex variable; these diﬀer from most of the literature, where the coeﬃcient is 1 in each deﬁnition. Spacetime actions are written as S and world-sheet actions as S . This presents a problem for D-branes, which are T -dual to the former and S -dual to the latter; S has been used arbitrarily. The spacetime metric is Gµν , while the world-sheet metric is γab (Minkowskian) or gab (Euclidean). In volume one, the spacetime Ricci tensor is Rµν and the world-sheet Ricci tensor is Rab . In volume two the former appears often and the latter never, so we have changed to Rµν for the spacetime Ricci tensor. xviii

Notation

xix

The following are used: ≡ ∼ = ≈ ∼

deﬁned as equivalent to approximately equal to equal up to nonsingular terms (in OPEs), or rough correspondence.

1 A ﬁrst look at strings

1.1

Why strings?

One of the main themes in the history of science has been uniﬁcation. Time and again diverse phenomena have been understood in terms of a small number of underlying principles and building blocks. The principle that underlies our current understanding of nature is quantum ﬁeld theory, quantum mechanics with the basic observables living at spacetime points. In the late 1940s it was shown that quantum ﬁeld theory is the correct framework for the uniﬁcation of quantum mechanics and electromagnetism. By the early 1970s it was understood that the weak and strong nuclear forces are also described by quantum ﬁeld theory. The full theory, the S U(3) × S U(2) × U(1) Model or Standard Model, has been conﬁrmed repeatedly in the ensuing years. Combined with general relativity, this theory is consistent with virtually all physics down to the scales probed by particle accelerators, roughly 10−16 cm. It also passes a variety of indirect tests that probe to shorter distances, including precision tests of quantum electrodynamics, searches for rare meson decays, limits on neutrino masses, limits on axions (light weakly interacting particles) from astrophysics, searches for proton decay, and gravitational limits on the couplings of massless scalars. In each of these indirect tests new physics might well have appeared, but in no case has clear evidence for it yet been seen; at the time of writing, the strongest sign is the solar neutrino problem, suggesting nonzero neutrino masses. The Standard Model (plus gravity) has a fairly simple structure. There are four interactions based on local invariance principles. One of these, gravitation, is mediated by the spin-2 graviton, while the other three are mediated by the spin-1 S U(3) × S U(2) × U(1) gauge bosons. In addition, the theory includes the spin-0 Higgs boson needed for symmetry breaking, and the quarks and leptons, ﬁfteen multiplets of spin- 12 fermions in three 1

2

1 A ﬁrst look at strings

generations of ﬁve. The dynamics is governed by a Lagrangian that depends upon roughly twenty free parameters such as the gauge and Yukawa couplings. In spite of its impressive successes, this theory is surely not complete. First, it is too arbitrary: why does this particular pattern of gauge ﬁelds and multiplets exist, and what determines the parameters in the Lagrangian? Second, the union of gravity with quantum theory yields a nonrenormalizable quantum ﬁeld theory, a strong signal that new physics should appear at very high energy. Third, even at the classical level the theory breaks down at the singularities of general relativity. Fourth, the theory is in a certain sense unnatural: some of the parameters in the Lagrangian are much smaller than one would expect them to be. It is these problems, rather than any positive experimental evidence, that presently must guide us in our attempts to ﬁnd a more complete theory. One seeks a principle that uniﬁes the ﬁelds of the Standard Model in a simpler structure, and resolves the divergence and naturalness problems. Several promising ideas have been put forward. One is grand uniﬁcation. This combines the three gauge interactions into one and the ﬁve multiplets of each generation into two or even one. It also successfully predicts one of the free parameters (the weak mixing angle) and possibly another (the bottom-tau mass ratio). A second idea is that spacetime has more than four dimensions, with the additional ones so highly curved as to be undetectable at current energies. This is certainly a logical possibility, since spacetime geometry is dynamical in general relativity. What makes it attractive is that a single higher-dimensional ﬁeld can give rise to many four-dimensional ﬁelds, diﬀering in their polarization (which can point along the small dimensions or the large) and in their dependence on the small dimensions. This opens the possibility of unifying the gauge interactions and gravity (the Kaluza–Klein mechanism). It also gives a natural mechanism for producing generations, repeated copies of the same fermion multiplets. A third unifying principle is supersymmetry, which relates ﬁelds of diﬀerent spins and statistics, and which helps with the divergence and naturalness problems. Each of these ideas — grand uniﬁcation, extra dimensions, and supersymmetry — has attractive features and is consistent with the various tests of the Standard Model. It is plausible that these will be found as elements of a more complete theory of fundamental physics. It is clear, however, that something is still missing. Applying these ideas, either singly or together, has not led to theories that are substantially simpler or less arbitrary than the Standard Model. Short-distance divergences have been an important issue many times in quantum ﬁeld theory. For example, they were a key clue leading from the Fermi theory of the weak interaction to the Weinberg–Salam theory. Let

3

1.1 Why strings?

(a)

(b)

(c)

Fig. 1.1. (a) Two particles propagating freely. (b) Correction from one-graviton exchange. (c) Correction from two-graviton exchange.

us look at the short-distance problem of quantum gravity, which can be understood from a little dimensional analysis. Figure 1.1 shows a process, two particles propagating, and corrections due to one-graviton exchange and two-graviton exchange. The one-graviton exchange is proportional to Newton’s constant GN . The ratio of the one-graviton correction to the original amplitude must be governed by the dimensionless combination GN E 2 ¯h−1 c−5 , where E is the characteristic energy of the process; this is the only dimensionless combination that can be formed from the parameters in the problem. Throughout this book we will use units in which h¯ = c = 1, deﬁning the Planck mass −1/2

MP = GN

= 1.22 × 1019 GeV

(1.1.1)

and the Planck length MP−1 = 1.6 × 10−33 cm .

(1.1.2)

The ratio of the one-graviton to the zero-graviton amplitude is then of order (E/MP )2 . From this dimensional analysis one learns immediately that the quantum gravitational correction is an irrelevant interaction, meaning that it grows weaker at low energy, and in particular is negligible at particle physics energies of hundreds of GeV. By the same token, the coupling grows stronger at high energy and at E > MP perturbation theory breaks down. In the two-graviton correction of ﬁgure 1.1(c) there is a sum over intermediate states. For intermediate states of high energy E , the ratio of the two-graviton to the zero-graviton amplitude is on dimensional grounds of order G2N E 2

dE E =

E2 MP4

dE E ,

(1.1.3)

4

1 A ﬁrst look at strings

which diverges if the theory is extrapolated to arbitrarily high energies. In position space this divergence comes from the limit where all the graviton vertices become coincident. The divergence grows worse with each additional graviton — this is the problem of nonrenormalizability. There are two possible resolutions. The ﬁrst is that the divergence is due to expanding in powers of the interaction and disappears when the theory is treated exactly. In the language of the renormalization group, this would be a nontrivial UV ﬁxed point. The second is that the extrapolation of the theory to arbitrarily high energies is incorrect, and beyond some energy the theory is modiﬁed in a way that smears out the interaction in spacetime and softens the divergence. It is not known whether quantum gravity has a nontrivial UV ﬁxed point, but there are a number of reasons for concentrating on the second possibility. One is history — the same kind of divergence problem in the Fermi theory of the weak interaction was a sign of new physics, the contact interaction between the fermions resolving at shorter distance into the exchange of a gauge boson. Another is that we need a more complete theory in any case to account for the patterns in the Standard Model, and it is reasonable to hope that the same new physics will solve the divergence problem of quantum gravity. In quantum ﬁeld theory it is not easy to smear out interactions in a way that preserves the consistency of the theory. We know that Lorentz invariance holds to very good approximation, and this means that if we spread the interaction in space we spread it in time as well, with consequent loss of causality or unitarity. Moreover we know that Lorentz invariance is actually embedded in a local symmetry, general coordinate invariance, and this makes it even harder to spread the interaction out without producing inconsistencies. In fact, there is presently only one way known to spread out the gravitational interaction and cut oﬀ the divergence without spoiling the consistency of the theory. This is string theory, illustrated in ﬁgure 1.2. In this theory the graviton and all other elementary particles are onedimensional objects, strings, rather than points as in quantum ﬁeld theory. Why this should work and not anything else is not at all obvious a priori, but as we develop the theory we will see how it comes about.1 Perhaps we merely suﬀer from a lack of imagination, and there are many other consistent theories of gravity with a short-distance cutoﬀ. However, experience has shown that divergence problems in quantum ﬁeld theory 1

There is an intuitive answer to at least one common question: why not membranes, two- or higher-dimensional objects? The answer is that as we spread out particles in more dimensions we reduce the spacetime divergences, but encounter new divergences coming from the increased number of internal degrees of freedom. One dimension appears to be the unique case where both the spacetime and internal divergences are under control. However, as we will discuss in chapter 14, the membrane idea has resurfaced in somewhat transmuted form as matrix theory.

1.1 Why strings?

5

(a)

(b)

(c)

Fig. 1.2. (a) Closed string. (b) Open string. (c) The loop amplitude of ﬁg. 1.1(c) in string theory. Each particle world-line becomes a cylinder, and the interactions no longer occur at points. (The cross-sections on the intermediate lines are included only for perspective.)

are not easily resolved, so if we have even one solution we should take it very seriously. Indeed, we are fortunate that consistency turns out to be such a restrictive principle, since the uniﬁcation of gravity with the other interactions takes place at such high energy, MP , that experimental tests will be diﬃcult and indirect. So what do we ﬁnd if we pursue this idea? In a word, the result is remarkable. String theory dovetails beautifully with the previous ideas for explaining the patterns in the Standard Model, and does so with a structure more elegant and uniﬁed than in quantum ﬁeld theory. In particular, if one tries to construct a consistent relativistic quantum theory of one-dimensional objects one ﬁnds: 1. Gravity. Every consistent string theory must contain a massless spin-2 state, whose interactions reduce at low energy to general relativity. 2. A consistent theory of quantum gravity, at least in perturbation theory. As we have noted, this is in contrast to all known quantum ﬁeld theories of gravity. 3. Grand uniﬁcation. String theories lead to gauge groups large enough to include the Standard Model. Some of the simplest string theories lead to the same gauge groups and fermion representations that arise in the uniﬁcation of the Standard Model.

6

1 A ﬁrst look at strings 4. Extra dimensions. String theory requires a deﬁnite number of spacetime dimensions, ten.2 The ﬁeld equations have solutions with four large ﬂat and six small curved dimensions, with four-dimensional physics that resembles the Standard Model. 5. Supersymmetry. Consistent string theories require spacetime supersymmetry, as either a manifest or a spontaneously broken symmetry. 6. Chiral gauge couplings. The gauge interactions in nature are parity asymmetric (chiral). This has been a stumbling block for a number of previous unifying ideas: they required parity symmetric gauge couplings. String theory allows chiral gauge couplings. 7. No free parameters. String theory has no adjustable constants. 8. Uniqueness. Not only are there no continuous parameters, but there is no discrete freedom analogous to the choice of gauge group and representations in ﬁeld theory: there is a unique string theory.

In addition one ﬁnds a number of other features, such as an axion, and hidden gauge groups, that have played a role in ideas for uniﬁcation. This is a remarkable list, springing from the simple supposition of onedimensional objects. The ﬁrst two points alone would be of great interest. The next four points come strikingly close to the picture one arrives at in trying to unify the Standard Model. And as indicated by the last two points, string theory accomplishes this with a structure that is tighter and less arbitrary than in quantum ﬁeld theory, supplying the element missing in the previous ideas. The ﬁrst point is a further example of this tightness: string theory must have a graviton, whereas in ﬁeld theory this and other ﬁelds are combined in a mix-and-match fashion. String theory further has connections to many areas of mathematics, and has led to the discovery of new and unexpected relations among them. It has rich connections to the recent discoveries in supersymmetric quantum ﬁeld theory. String theory has also begun to address some of the deeper questions of quantum gravity, in particular the quantum mechanics of black holes. Of course, much remains to be done. String theory may resemble the real world in its broad outlines, but a decisive test still seems to be far away. The main problem is that while there is a unique theory, it has an enormous number of classical solutions, even if we restrict attention 2

To be precise, string theory modiﬁes the notions of spacetime topology and geometry, so what we mean by a dimension here is generalized. Also, we will see that ten dimensions is the appropriate number for weakly coupled string theory, but that the picture can change at strong coupling.

1.1 Why strings?

7

to solutions with four large ﬂat dimensions. Upon quantization, each of these is a possible ground state (vacuum) for the theory, and the fourdimensional physics is diﬀerent in each. It is known that quantum eﬀects greatly reduce the number of stable solutions, but a full understanding of the dynamics is not yet in hand. It is worth recalling that even in the Standard Model, the dynamics of the vacuum plays an important role in the physics we see. In the electroweak interaction, the fact that the vacuum is less symmetric than the Hamiltonian (spontaneous symmetry breaking) plays a central role. In the strong interaction, large ﬂuctuating gauge ﬁelds in the vacuum are responsible for quark conﬁnement. These phenomena in quantum ﬁeld theory arise from having a quantum system with many degrees of freedom. In string theory there are seemingly many more degrees of freedom, and so we should expect even richer dynamics. Beyond this, there is the question, ‘what is string theory?’ Until recently our understanding of string theory was limited to perturbation theory, small numbers of strings interacting weakly. It was not known how even to deﬁne the theory at strong coupling. There has been a suspicion that the degrees of freedom that we use at weak coupling, one-dimensional objects, are not ultimately the simplest or most complete way to understand the theory. In the past few years there has been a great deal of progress on these issues, growing largely out of the systematic application of the constraints imposed by supersymmetry. We certainly do not have a complete understanding of the dynamics of strongly coupled strings, but it has become possible to map out in detail the space of vacua (when there is enough unbroken supersymmetry) and this has led to many surprises. One is the absolute uniqueness of the theory: whereas there are several weakly coupled string theories, all turn out to be limits in the space of vacua of a single theory. Another is a limit in which spacetime becomes elevendimensional, an interesting number from the point of view of supergravity but impossible in weakly coupled string theory. It has also been understood that the theory contains new extended objects, D-branes, and this has led to the new understanding of black hole quantum mechanics. All this also gives new and unexpected clues as to the ultimate nature of the theory. In summary, we are fortunate that so many approaches seem to converge on a single compelling idea. Whether one starts with the divergence problem of quantum gravity, with attempts to account for the patterns in the Standard Model, or with a search for new symmetries or mathematical structures that may be useful in constructing a uniﬁed theory, one is led to string theory.

8

1 A ﬁrst look at strings Outline

The goal of these two volumes is to provide a complete introduction to string theory, starting at the beginning and proceeding through the compactiﬁcation to four dimensions and to the latest developments in strongly coupled strings. Volume one is an introduction to bosonic string theory. This is not a realistic theory — it does not have fermions, and as far as is known has no stable ground state. The philosophy here is the same as in starting a course on quantum ﬁeld theory with a thorough study of scalar ﬁeld theory. That is also not the theory one is ultimately interested in, but it provides a simple example for developing the unique dynamical and technical features of quantum ﬁeld theory before introducing the complications of spin and gauge invariance. Similarly, a thorough study of bosonic string theory will give us a framework to which we can in short order add the additional complications of fermions and supersymmetry. The rest of chapter 1 is introductory. We present ﬁrst the action principle for the dynamics of string. We then carry out a quick and heuristic quantization using light-cone gauge, to show the reader some of the important aspects of the string spectrum. Chapters 2–7 are the basic introduction to bosonic string theory. Chapter 2 introduces the needed technical tools in the world-sheet quantum ﬁeld theory, such as conformal invariance, the operator product expansion, and vertex operators. Chapters 3 and 4 carry out the covariant quantization of the string, starting from the Polyakov path integral. Chapters 5–7 treat interactions, presenting the general formalism and applying it to tree-level and one-loop amplitudes. Chapter 8 treats the simplest compactiﬁcation of string theory, making some of the dimensions periodic. In addition to the phenomena that arise in compactiﬁed ﬁeld theory, such as Kaluza–Klein gauge symmetry, there is also a great deal of ‘stringy’ physics, including enhanced gauge symmetries, T -duality, and D-branes. Chapter 9 treats higher order amplitudes. The ﬁrst half outlines the argument that string theory in perturbation theory is ﬁnite and unitary as advertised; the second half presents brief treatments of a number of advanced topics, such as string ﬁeld theory. Appendix A is an introduction to path integration, so that our use of quantum ﬁeld theory is self-contained. Volume two treats supersymmetric string theories, focusing ﬁrst on ten-dimensional and other highly symmetric vacua, and then on realistic four-dimensional vacua. In chapters 10–12 we extend the earlier introduction to the supersymmetric string theories, developing the type I, II, and heterotic superstrings and their interactions. We then introduce the latest results in these subjects. Chapter 13 develops the properties and dynamics of D-branes, still

1.2 Action principles

9

using the tools of string perturbation theory as developed earlier in the book. Chapter 14 then uses arguments based on supersymmetry to understand strongly coupled strings. We ﬁnd that the strongly coupled limit of any string theory is described by a dual weakly coupled string theory, or by a new eleven-dimensional theory known provisionally as M-theory. We discuss the status of the search for a complete formulation of string theory and present one promising idea, m(atrix) theory. We brieﬂy discuss the quantum mechanics of black holes, carrying out the simplest entropy calculation. Chapter 15 collects a number of advanced applications of the various world-sheet symmetry algebras. Chapters 16 and 17 present four-dimensional string theories based on orbifold and Calabi–Yau compactiﬁcations. The goal is not an exhaustive treatment but rather to make contact between the simplest examples and the uniﬁcation of the Standard Model. Chapter 18 collects results that hold in wide classes of string theories, using general arguments based on worldsheet and spacetime gauge symmetries. Chapter 19 consists of advanced topics, including (2,2) world-sheet supersymmetry, mirror symmetry, the conifold transition, and the strong-coupling behavior of some compactiﬁed theories. Annotated reference lists appear at the end of each volume. I have tried to assemble a selection of articles, particularly reviews, that may be useful to the student. A glossary also appears at the end of each volume. 1.2

Action principles

We want to study the classical and quantum mechanics of a one-dimensional object, a string. The string moves in D ﬂat spacetime dimensions, with metric ηµν = diag(−, +, +, · · · , +). It is useful to review ﬁrst the classical mechanics of a zero-dimensional object, a relativistic point particle. We can describe the motion of a particle by giving its position in terms of D−1 functions of time, X(X 0 ). This hides the covariance of the theory though, so it is better to introduce a parameter τ along the particle’s world-line and describe the motion in spacetime by D functions X µ (τ). The parameterization is arbitrary: a diﬀerent parameterization of the same path is physically equivalent, and all physical quantities must be independent of this choice. That is, for any monotonic function τ (τ), the two paths X µ and X µ are the same, where X µ (τ (τ)) = X µ (τ) .

(1.2.1)

We are trading a less symmetric description for a more symmetric but redundant one, which is often a useful step. Figure 1.3(a) shows a parameterized world-line.

10

1 A ﬁrst look at strings

X0

X0

τ =2

τ =2 τ =1

τ =1 τ= 0 τ=−1 τ =−2

τ= 0 τ=−1 τ =−2

σ =0

σ =l

(b)

(a)

X

X

Fig. 1.3. (a) Parameterized world-line of a point particle. (b) Parameterized world-sheet of an open string.

The simplest Poincar´e-invariant action that does not depend on the parameterization would be proportional to the proper time along the world-line, Spp = −m

˙ µX ˙ µ )1/2 , dτ (−X

(1.2.2)

where a dot denotes a τ-derivative. The variation of the action, after an integration by parts, is δSpp = −m

dτ ˙uµ δX µ ,

(1.2.3)

where ˙ ν )−1/2 ˙ µ (−X ˙ νX uµ = X

(1.2.4)

is the normalized D-velocity. The equation of motion ˙uµ = 0 thus describes free motion. The normalization constant m is the particle’s mass, as can be checked by looking at the nonrelativistic limit (exercise 1.1). The action can be put in another useful form by introducing an additional ﬁeld on the world-line, an independent world-line metric γττ (τ). It will be convenient to work with the tetrad η(τ) = (−γττ (τ))1/2 , which is deﬁned to be positive. We use the general relativity term tetrad in any number of dimensions, even though its root means ‘four.’ Then

1 ˙ µX ˙ µ − ηm2 . dτ η −1 X (1.2.5) 2 This action has the same symmetries as the earlier action Spp , namely Poincar´e invariance and world-line reparameterization invariance. Under Spp =

1.2 Action principles

11

the latter, η(τ) transforms as η (τ )dτ = η(τ)dτ .

(1.2.6)

The equation of motion from varying the tetrad is ˙ µ /m2 . ˙ µX η 2 = −X

(1.2.7)

becomes the earlier S . InciUsing this to eliminate η(τ), the action Spp pp dentally, Spp makes perfect sense for massless particles, while Spp does not work in that case. are equivalent classically, it is hard to Although the actions Spp and Spp make sense of Spp in a path integral because of its complicated form, with is quadratic in derivatives inside the square root. On the other hand Spp derivatives and its path integral is fairly easily evaluated, as we will see in later chapters. Presumably, any attempt to deﬁne the quantum theory for path integral, and we will Spp will lead to a result equivalent to the Spp take the latter as our starting point in deﬁning the quantum theory. A one-dimensional object will sweep out a two-dimensional worldsheet, which can be described in terms of two parameters, X µ (τ, σ) as in ﬁgure 1.3(b). As in the case of the particle, we insist that physical quantities such as the action depend only on the embedding in spacetime and not on the parameterization. Not only is it attractive that the theory should have this property, but we will see that it is necessary in a consistent relativistic quantum theory. The simplest invariant action, the Nambu–Goto action, is proportional to the area of the world-sheet. In order to express this action in terms of X µ (τ, σ), deﬁne ﬁrst the induced metric hab where indices a, b, . . . run over values (τ, σ):

hab = ∂a X µ ∂b Xµ .

(1.2.8)

Then the Nambu–Goto action is

SNG =

M

dτdσ LNG ,

(1.2.9a)

1 (− det hab )1/2 , (1.2.9b) 2πα where M denotes the world-sheet. The constant α , which has units of spacetime-length-squared, is the Regge slope, whose signiﬁcance will be seen later. The tension T of the string is related to the Regge slope by LNG = −

T =

1 2πα

(1.2.10)

(exercise 1.1). Let us now consider the symmetries of the action, transformations of X µ (τ, σ) such that SNG [X ] = SNG [X]. These are:

12

1 A ﬁrst look at strings 1. The isometry group of ﬂat spacetime, the D-dimensional Poincar´e group, X µ (τ, σ) = Λµν X ν (τ, σ) + aµ

(1.2.11)

with Λµν a Lorentz transformation and aµ a translation. 2. Two-dimensional coordinate invariance, often called diﬀeomorphism (diﬀ) invariance. For new coordinates (τ (τ, σ), σ (τ, σ)), the transformation is X µ (τ , σ ) = X µ (τ, σ) .

(1.2.12)

The Nambu–Goto action is analogous to the point particle action Spp , with derivatives in the square root. Again we can simplify it by introducing an independent world-sheet metric γab (τ, σ). Henceforth ‘metric’ will always mean γab (unless we specify ‘induced’), and indices will always be raised and lowered with this metric. We will take γab to have Lorentzian signature (−, +). The action is

1 dτ dσ (−γ)1/2 γ ab ∂a X µ ∂b Xµ , (1.2.13) 4πα M where γ = det γab . This is the Brink–Di Vecchia–Howe–Deser–Zumino action, or Polyakov action for short. It was found by Brink, Di Vecchia, and Howe and by Deser and Zumino, in the course of deriving a generalization with local world-sheet supersymmetry. Its virtues, especially for path integral quantization, were emphasized by Polyakov. To see the equivalence to SNG , use the equation of motion obtained by varying the metric, SP [X, γ] = −

1 1/2 ab cd 1 dτ dσ (−γ) δγ h − γ γ h (1.2.14) ab cd , 2 ab 4πα M where hab is again the induced metric (1.2.8). We have used the general relation for the variation of a determinant,

δγ SP [X, γ] = −

δγ = γγ ab δγab = −γγab δγ ab .

(1.2.15)

hab = 12 γab γ cd hcd .

(1.2.16)

Then δγ SP = 0 implies

Dividing this equation by the square root of minus its determinant gives hab (−h)−1/2 = γab (−γ)−1/2 ,

(1.2.17)

so that γab is proportional to the induced metric. This in turn can be used to eliminate γab from the action, SP [X, γ] → −

1 2πα

dτ dσ (−h)1/2 = SNG [X] .

(1.2.18)

1.2 Action principles

13

The action SP has the following symmetries: 1. D-dimensional Poincar´e invariance: X (τ, σ) = Λµν X ν (τ, σ) + aµ , γab (τ, σ) = γab (τ, σ) . µ

(1.2.19)

2. Diﬀ invariance: X (τ , σ ) = X µ (τ, σ) , µ

∂σ c ∂σ d γ (τ , σ ) = γab (τ, σ) , ∂σ a ∂σ b cd for new coordinates σ a (τ, σ).

(1.2.20)

3. Two-dimensional Weyl invariance: X (τ, σ) = X µ (τ, σ) , γab (τ, σ) = exp(2ω(τ, σ)) γab (τ, σ) , µ

(1.2.21)

for arbitrary ω(τ, σ) . The Weyl invariance, a local rescaling of the world-sheet metric, has no analog in the Nambu–Goto form. We can understand its appearance by looking back at the equation of motion (1.2.17) used to relate the Polyakov and Nambu–Goto actions. This does not determine γab completely but only up to a local rescaling, so Weyl-equivalent metrics correspond to the same embedding in spacetime. This is an extra redundancy of the Polyakov formulation, in addition to the obvious diﬀ invariance. The variation of the action with respect to the metric deﬁnes the energymomentum tensor, T ab (τ, σ) = −4π(−γ)−1/2

δ SP δγab

1 = − (∂a X µ ∂b Xµ − 12 γ ab ∂c X µ ∂c Xµ ) . (1.2.22) α This has an extra factor of −2π relative to the usual deﬁnition in ﬁeld theory; this is a standard and convenient convention in string theory. It is conserved, ∇a T ab = 0, as a consequence of diﬀ invariance. The invariance of SP under arbitrary Weyl transformations further implies that γab

δ SP = 0 ⇒ Taa = 0 . δγab

(1.2.23)

The actions SNG and SP deﬁne two-dimensional ﬁeld theories on the string world-sheet. In string theory, we will see that amplitudes for spacetime processes are given by matrix elements in the two-dimensional quantum ﬁeld theory on the world-sheet. While our interest is four-dimensional

14

1 A ﬁrst look at strings

spacetime, most of the machinery we use in string perturbation theory is two-dimensional. From the point of view of the world-sheet, the coordinate transformation law (1.2.12) deﬁnes X µ (τ, σ) as a scalar ﬁeld, with µ going along for the ride as an internal index. From the two-dimensional point of view the Polyakov action describes massless Klein–Gordon scalars X µ covariantly coupled to the metric γab . Also from this point of view, the Poincar´e invariance is an internal symmetry, meaning that it acts on ﬁelds at ﬁxed τ, σ. Varying γab in the action gives the equation of motion Tab = 0 .

(1.2.24)

Varying X µ gives the equation of motion ∂a [(−γ)1/2 γ ab ∂b X µ ] = (−γ)1/2 ∇2 X µ = 0 .

(1.2.25)

For world-sheets with boundary there is also a surface term in the variation of the action. To be speciﬁc, take the coordinate region to be −∞<τ<∞ ,

0≤σ≤.

(1.2.26)

We will think of τ as a time variable and σ as spatial, so this is a single string propagating without sources. Then 1 δSP = 2πα

∞

−∞

dτ 0

dσ (−γ)1/2 δX µ ∇2 Xµ

σ= ∞ 1 1/2 µ σ − dτ (−γ) δX ∂ X . (1.2.27) µ σ=0 2πα −∞ In the interior, the equation of motion is as above. The boundary term vanishes if

∂σ X µ (τ, 0) = ∂σ X µ (τ, ) = 0 . These are Neumann boundary conditions on na ∂a X µ = 0

X µ.

(1.2.28)

Stated more covariantly,

on ∂M ,

(1.2.29)

where na is normal to the boundary ∂M. The ends of the open string move freely in spacetime. The surface term in the equation of motion will also vanish if we impose X µ (τ, ) = X µ (τ, 0) , γab (τ, ) = γab (τ, 0) .

∂σ X µ (τ, ) = ∂σ X µ (τ, 0) ,

(1.2.30a) (1.2.30b)

That is, the ﬁelds are periodic. There is no boundary; the endpoints are joined to form a closed loop. The open string boundary condition (1.2.28) and closed string boundary condition (1.2.30) are the only possibilities consistent with D-dimensional

15

1.2 Action principles

Poincar´e invariance and the equations of motion. If we relax the condition of Poincar´e invariance then there are other possibilities which will be important later. Some of these are explored in the exercises. The Nambu–Goto and Polyakov actions may be the simplest with the given symmetries, but simplicity of the action is not the right criterion in quantum ﬁeld theory. Symmetry, not simplicity, is the key idea: for both physical and technical reasons we must usually consider the most general local action consistent with all of the symmetries of the theory. Let us try to generalize the Polyakov action, requiring that the symmetries be maintained and that the action be polynomial in derivatives (else it is hard to make sense of the quantum theory).3 Global Weyl invariance, ω(τ, σ) = constant, requires that the action have one more factor of γ ab than γab to cancel the variation of (−γ)1/2 . The extra upper indices can only contract with derivatives, so each term has precisely two derivatives. The coordinate invariance and Poincar´e invariance then allow one additional term beyond the original SP :

1 χ= dτ dσ (−γ)1/2 R , (1.2.31) 4π M where R is the Ricci scalar constructed from γab . Under a local Weyl rescaling, (−γ )1/2 R = (−γ)1/2 (R − 2∇2 ω) .

(1.2.32)

The variation is a total derivative, because (−γ)1/2 ∇a v a = ∂a ((−γ)1/2 v a ) for any v a . The integral (1.2.31) is therefore invariant for a world-sheet without boundary. With boundaries an additional surface term is needed (exercise 1.3). Since χ is allowed by the symmetries we will include it in the action: SP = SP − λχ . =−

dτ dσ (−γ) M

1/2

1 ab λ γ ∂a X µ ∂b X µ + R 4πα 4π

.

(1.2.33)

This is the most general (diﬀ×Weyl)-invariant and Poincar´e-invariant action with these ﬁelds and symmetries. For the moment the discussion of the symmetries is classical — we are ignoring possible quantum anomalies, which will be considered in chapter 3. The action SP looks like the Hilbert action for the metric, (−γ)1/2 R, minimally coupled to D massless scalar ﬁelds X µ . However, in two dimen3

If we relax this requirement then there are various additional possibilities, such as invariants constructed from the curvature of the induced metric hab . These would be expected to appear for such one-dimensional objects as electric and magnetic ﬂux tubes, and in that case the path integral makes sense because the thickness of the object provides a cutoﬀ. But for inﬁnitely thin strings, which is what we have in fundamental string theory, this is not possible.

16

1 A ﬁrst look at strings

sions, the Hilbert action depends only on the topology of the world-sheet and does not give dynamics to the metric. To see this, we use the result from general relativity that its variation is proportional to Rab − 12 γab R. In two dimensions, the symmetries of the curvature tensor imply that Rab = 12 γab R and so this vanishes: the Hilbert action is invariant under any continuous change in the metric. It does have some global signiﬁcance, as we will see in the chapter 3. There are various further ways one might try to generalize the theory. One, suggested by our eventual interest in D = 4, is to allow more than four X µ ﬁelds but require only four-dimensional Poincar´e invariance. A second, motivated by the idea that symmetry is paramount, is to keep the same local symmetries plus four-dimensional Poincar´e invariance, but to allow a more general ﬁeld content — that is, two-dimensional ﬁelds in diﬀerent representations of the world-sheet and spacetime symmetry groups. These ideas are both important and we will return to them at various times. A further idea is to enlarge the local symmetry of the theory; we will pursue this idea in volume two. 1.3

The open string spectrum

We discuss the spectrum of the open string in this section and that of the closed in the next, using light-cone gauge to eliminate the diﬀ×Weyl redundancy. This gauge hides the covariance of the theory, but it is the quickest route to the spectrum and reveals important features like the critical dimension and the existence of massless gauge particles. The emphasis here is on the results, not the method, so the reader should not dwell on ﬁne points. A systematic treatment using the covariant conformal gauge will begin in the next chapter. Deﬁne light-cone coordinates in spacetime: x± = 2−1/2 (x0 ± x1 ) ,

xi , i = 2, . . . , D − 1 .

(1.3.1)

We are using lower case for the spacetime coordinates xµ and upper case for the associated world-sheet ﬁelds X µ (σ, τ). In these coordinates the metric is aµ bµ = −a+ b− − a− b+ + ai bi , a− = −a , +

−

a+ = −a ,

(1.3.2a) i

ai = a .

(1.3.2b)

We will set the world-sheet parameter τ at each point of the world-sheet to be equal to the spacetime coordinate x+ . One might have thought to try τ = x0 but this does not lead to the same simpliﬁcations. So x+ will play the role of time and p− that of energy. The longitudinal variables x− and

1.3 The open string spectrum

17

p+ are then like spatial coordinates and momenta, as are the transverse xi and pi . We start by illustrating the procedure in the case of the point particle, . Fix the parameterization of the world-line by using the action Spp X + (τ) = τ .

(1.3.3)

The action then becomes 1 ˙ − + η −1 X ˙ iX ˙ i − ηm2 . Spp = dτ −2η −1 X 2 ˙ µ are The canonical momenta pµ = ∂L/∂X

(1.3.4)

p− = −η −1 ,

(1.3.5)

˙i . pi = η −1 X

Also recall the metric pi = pi and p+ = −p− . The Hamiltonian is ˙ − + pi X ˙i − L H = p− X pi pi + m2 . (1.3.6) 2p+ ˙ + because X + is not a dynamical variable Note that there is no term p+ X in the gauge-ﬁxed theory. Also, η˙ does not appear in the action and so the momentum pη vanishes. Since η = −1/p− it is clear that we should not treat η as an independent canonical coordinate; in a more complete treatment we would justify this by appealing to the theory of systems with constraints. To quantize, impose canonical commutators on the dynamical ﬁelds, =

[pi , X j ] = −iδi j ,

[p− , X − ] = −i .

(1.3.7)

The momentum eigenstates |k− form a complete set. The remaining momentum component p+ is determined in terms of the others as follows. The gauge choice relates τ and X + translations, so H = −p+ = p− . The relative sign between H and p+ arises because the former is active and the latter passive. Then eq. (1.3.6) becomes the relativistic mass-shell condition, and we have obtained the spectrum of a relativistic scalar. Now turn to the open string, taking again the coordinate region −∞ < τ < ∞ and 0 ≤ σ ≤ . Again we must make a gauge choice to ﬁx the redundancies in the Polyakov action, and by a good choice we can also make the equations of motion simple. Set , ki

X+ = τ , ∂σ γσσ = 0 , det γab = −1 .

(1.3.8a) (1.3.8b) (1.3.8c)

That is, we choose light-cone gauge for the world-sheet time coordinate and impose two conditions on the metric. This is three conditions for

18

1 A ﬁrst look at strings

three local symmetries (the choice of two world-sheet coordinates plus the Weyl scaling). Let us see how to reach the gauge (1.3.8). First choose τ according to (1.3.8a). Now notice that f = γσσ (− det γab )−1/2 transforms as f dσ = fdσ

(1.3.9)

under reparameterizations of σ where τ is left ﬁxed. Thus we deﬁne an invariant length fdσ = dl. Deﬁne the σ coordinate of a point to be proportional to its invariant distance dl from the σ = 0 endpoint; the constant of proportionality is determined by requiring that the coordinate of the right-hand endpoint remain at σ = . In this coordinate system f = dl/dσ is independent of σ. It still depends on τ; we would like the parameter region 0 ≤ σ ≤ to remain unchanged, so there is not enough freedom to remove the τ-dependence. Finally, make a Weyl transformation to satisfy condition (1.3.8c). Since f is Weyl-invariant, ∂σ f still vanishes. With condition (1.3.8c) this implies that ∂σ γσσ = 0, and so condition (1.3.8b) is satisﬁed as well.4 We can solve the gauge condition (1.3.8c) for γττ (τ, σ), and γσσ is independent of σ, so the independent degrees of freedom in the metric are now γσσ (τ) and γστ (τ, σ). In terms of these, the inverse metric is

γ ττ γ τσ γ στ γ σσ

=

−γσσ (τ) γτσ (τ, σ) −1 (τ)(1 − γ 2 (τ, σ)) γτσ (τ, σ) γσσ τσ

.

(1.3.10)

The Polyakov Lagrangian then becomes 1 L=− 4πα

0

dσ γσσ (2∂τ x− − ∂τ X i ∂τ X i )

−1 2 −2γστ (∂σ Y − − ∂τ X i ∂σ X i ) + γσσ (1 − γτσ )∂σ X i ∂σ X i .

(1.3.11) Here we have separated X − (τ, σ) into two pieces x− (τ) and Y − (τ, σ), the ﬁrst being the mean value of X − at given τ and the second having a mean value of zero. That is,

1 dσX − (τ, σ) , x (τ) = 0 Y − (τ, σ) = X − (τ, σ) − x− (τ) . −

4

(1.3.12a) (1.3.12b)

We have shown how to assign a deﬁnite (τ, σ) value to each point on the world-sheet, but we have not shown that this is a good gauge — that diﬀerent points have diﬀerent coordinates. Already for the point particle this is an issue: τ is a good coordinate only if the world-line does not double back through an x+ -hyperplane. For the string we need this and also that f is positive everywhere. It is not hard to show that for allowed classical motions, in which all points travel at less than the speed of light, these conditions are met. In the quantum theory, one can show for the point particle that the doubled-back world-lines give no net contribution. Presumably this can be generalized to the string, but this is one of those ﬁne points not to dwell on.

19

1.3 The open string spectrum

The ﬁeld Y − does not appear in any terms with time derivatives and so is nondynamical. It acts as a Lagrange multiplier, constraining ∂σ2 γτσ to vanish (extra ∂σ because Y − has zero mean). Note further that in the gauge (1.3.8) the open string boundary condition (1.2.28) becomes γτσ ∂τ X µ − γττ ∂σ X µ = 0

at σ = 0, .

(1.3.13)

For µ = + this gives γτσ = 0

at σ = 0,

(1.3.14)

and since ∂σ2 γτσ = 0, then γτσ vanishes everywhere. For µ = i, the boundary condition is ∂σ X i = 0

at σ = 0, .

(1.3.15)

After imposing the gauge conditions and taking into account the Lagrange multiplier Y − , we can proceed by ordinary canonical methods. The system is reduced to the variables x− (τ), γσσ (τ) and the ﬁelds X i (τ, σ). The Lagrangian is

1 − i i −1 i i γ ∂ x + dσ γ ∂ X ∂ X − γ ∂ X ∂ X . (1.3.16) σσ τ σσ τ τ σ σσ σ 2πα 4πα 0 The momentum conjugate to x− is

L=−

p− = −p+ =

∂L γσσ . =− − ∂(∂τ x ) 2πα

(1.3.17)

As with η in the particle example, γσσ is a momentum and not a coordinate. The momentum density conjugate to X i (τ, σ) is Πi =

δL p+ 1 i γ ∂ X = = ∂τ X i . σσ τ δ(∂τ X i ) 2πα

The Hamiltonian is then −

H = p− ∂τ x − L + = 4πα p+

0

0

(1.3.18)

dσ Πi ∂τ X i

1 dσ 2πα Π Π + ∂σ X i ∂σ X i 2πα

i

i

.

(1.3.19)

This is just the Hamiltonian for D − 2 free ﬁelds X i , with p+ a conserved quantity. The equations of motion are ∂H H ∂H = + , ∂ τ p+ = − = 0 , ∂p− p ∂x δH δH c 2 i = 2πα cΠi , ∂τ Πi = − i = ∂ X , ∂τ X i = δΠi δX 2πα σ implying the wave equation ∂τ x− =

∂τ2 X i = c2 ∂σ2 X i

(1.3.20a) (1.3.20b)

(1.3.21)

20

1 A ﬁrst look at strings

with velocity c = /2πα p+ . We will not develop the interactions in the light-cone formalism, but for these it is useful to choose the coordinate length for each string to be proportional to p+ , so that c = 1. Since p+ is positive and conserved, the total string length is then also conserved. The transverse coordinates satisfy a free wave equation so it is useful to expand in normal modes. In fact X ± also satisfy the free wave equation. For X + this is trivial, while for X − it requires a short calculation. The general solution to the wave equation with boundary condition (1.3.15) is

∞ pi πincτ πnσ 1 i αn exp − . (1.3.22) cos X (τ, σ) = x + + τ + i(2α )1/2 p n=−∞ n i

i

n=0

Reality of X i requires αi−n = (αin )† . We have deﬁned the center-of-mass variables 1 i x (τ) = dσ X i (τ, σ) , (1.3.23a) 0 p+ dσ Πi (τ, σ) = dσ ∂τ X i (τ, σ) , (1.3.23b) pi (τ) = 0 0 which are the average position and the total momentum. These are the Heisenberg operators; the Schr¨ odinger operators xi ≡ xi (0) and pi ≡ pi (0) appear in the expansion (1.3.22). To quantize, impose the equal time canonical commutation relations [x− , p+ ] = iη −+ = −i , [X i (σ), Πj (σ )] = iδ ij δ(σ − σ ) ,

(1.3.24a) (1.3.24b)

with all other commutators between the independent variables vanishing. In terms of the Fourier components, [xi , pj ] = iδ ij , [αim , αjn ] = mδ ij δm,−n .

(1.3.25a) (1.3.25b)

For each m and i the modes satisfy a harmonic oscillator algebra with nonstandard normalization αim ∼ m1/2 a ,

αi−m ∼ m1/2 a† ,

m>0,

(1.3.26)

where [a, a† ] = 1. The oscillators are labeled by the direction of oscillation i and the harmonic m. The state |0; k , where k = (k + , k i ), is deﬁned to be annihilated by the lowering operators and to be an eigenstate of the center-of-mass momenta, p+ |0; k = k + |0; k , pi |0; k = k i |0; k , αim |0; k = 0 , m > 0 .

(1.3.27a) (1.3.27b)

21

1.3 The open string spectrum

A general state can be built by acting on |0; k with the raising operators, |N; k =

D−1 ∞ i=2

(αi−n )Nin |0; k . (nNin Nin !)1/2 n=1

(1.3.28)

That is, the independent states can be labeled by the center-of-mass momenta k + and k i , and by the occupation numbers Nin for each mode (i, n), where i = 2, . . . , D − 1 and n = 1, . . . , ∞. The center-of-mass momenta are just the degrees of freedom of a point particle, while the oscillators represent an inﬁnite number of internal degrees of freedom. Every choice of these occupation numbers corresponds, from the spacetime point of view, to a diﬀerent particle or spin state. Note that the states (1.3.28) form the Hilbert space H1 of a single open string, and in particular the state |0; 0 is the ground state of a single string with zero momentum, not the zero-string vacuum state. We will call the latter |vacuum to make this clear. The various operators appearing above do not create or destroy strings but act within the space of states of a single string. The n-string Hilbert space Hn would be formed as the product of n copies of the space (1.3.28); the wavefunction must be symmetrized because, as we will see, all these states have integer spin. The full Hilbert space of the string theory, at least in the free limit that we are considering here, is then the sum H = |vacuum ⊕ H1 ⊕ H2 ⊕ . . . .

(1.3.29)

Inserting the mode expansion (1.3.22) into the Hamiltonian (1.3.19) gives pi p i 1 H= ++ + 2p 2p α

∞

αi−n αin

+A

.

(1.3.30)

n=1

In the Hamiltonian H, the order of operators is ambiguous. We have put the lowering operators on the right and the raising operators on the left, and included an unknown constant A from the commutators. In a careful treatment of light-cone quantization, this constant is determined as follows. The choice of light-cone gauge has obscured the Lorentz invariance of the theory. It is necessary to check this by ﬁnding the operators M µν that generate Lorentz transformations, and verifying that they have the correct algebra with pµ and with each other. One ﬁnds that this is the case only for the speciﬁc value A = −1, and only if the spacetime dimensionality is precisely D = 26. We do not wish to spend this much time on light-cone quantization, so we will try to show that the result is plausible without a systematic treatment. We assure the reader that we will obtain the values of A and D honestly in the conformal gauge approach, which will also give us insight

22

1 A ﬁrst look at strings

into why Lorentz invariance would be lost in the light-cone approach for the wrong A or D. First, we assert that the operator ordering constant in the Hamiltonian for a free ﬁeld comes from summing the zero-point energies of each oscillator mode, 12 ω for a bosonic ﬁeld like X µ . Equivalently, it always works out that the natural operator order is averaged, 12 ω(aa† + a† a), which is the same as ω(a† a + 12 ). In H this would give A=

∞ D−2 n, 2 n=1

(1.3.31)

the factor of D − 2 coming from the sum over transverse directions. The zero-point sum diverges. It can be evaluated by regulating the theory and then being careful to preserve Lorentz invariance in the renormalization. This leads to the odd result ∞

n→−

n=1

1 . 12

(1.3.32)

To motivate this, insert a smooth cutoﬀ factor −1/2 exp(−γσσ |kσ |)

(1.3.33) −1/2

into the sum, where kσ = nπ/ and the factor of γσσ is included to make this invariant under σ reparameterizations. The zero-point constant is then A→

∞

D−2 n exp −n(π/2p+ α )1/2 2 n=1

D − 2 2p+ α 1 = − + O() 2 2 π 12

(1.3.34)

.

The cutoﬀ-dependent ﬁrst term is proportional to the length of the and can be canceled by a counterterm in the action proportional to string 2 d σ(−γ)1/2 . In fact, Weyl invariance requires that it be canceled, leaving only the cutoﬀ-independent second term, A=

2−D . 24

(1.3.35)

This ﬁnite remainder is an example of a Casimir energy, coming from the fact that the string has a ﬁnite length. As for the point particle, p− = H, so

1 2−D m = 2p H − p p = N + α 24 2

+

i i

,

(1.3.36)

1.3 The open string spectrum

23

where N is the level N=

D−1 ∞

nNin .

(1.3.37)

i=2 n=1

The mass of each state is thus determined in terms of the level of excitation. Now let us look at some of the light string states. The lightest is 2−D . (1.3.38) 24α The mass-squared is negative if D > 2: the state is a tachyon. In ﬁeld theory the potential energy for a scalar ﬁeld is 12 m2 φ2 , so the negative mass-squared means that the no-string ‘vacuum’ is actually unstable, like the symmetric state in a spontaneously broken theory. It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known. Starting with our study of the superstring in volume two, we will see that there are tachyon-free string theories. For now, the simplest way to proceed is to ignore this instability and use the bosonic string theory as our model for developing string technology. Occasionally we will encounter divergences due to the tachyon, but these will not interfere with our purpose. The lowest excited states of the string are obtained by exciting one of the n = 1 modes once: 26 − D . (1.3.39) αi−1 |0; k , m2 = 24α Lorentz invariance now requires a speciﬁc value of D as follows. The analysis of spin is diﬀerent for massive and massless particles. For a massive particle, one goes to the rest frame pµ = (m, 0, . . . , 0). The internal states then form a representation of the spatial rotation group S O(D − 1). For a massless particle there is no rest frame; choose the frame pµ = (E, E, 0, . . . , 0). The S O(D − 2) acting on the transverse directions leaves pµ invariant, and the internal states form a representation of this smaller group. This is familiar from D = 4: massive particles are labeled by spin j, the S O(3) representation, and so have 2j + 1 states. Massless particles are labeled by their helicity λ, which is their eigenvalue under the single generator of S O(2). Lorentz invariance alone thus requires only one state, though CP T symmetry takes λ to −λ and so requires two states for λ = 0. In D dimensions, a massive vector particle thus has D − 1 spin states while a massless vector need have only D − 2 states. At the ﬁrst level we found only the D − 2 states αi−1 |0; k , so these must be massless and |0; k ,

m2 =

A = −1,

D = 26 .

(1.3.40)

This is a striking and important result: the spectrum is Lorentz-invariant only if the number of spacetime dimensions is D = 26. The classical theory

24

1 A ﬁrst look at strings

is Lorentz-invariant for any D, but there is an anomaly — the symmetry is not preserved by the quantization except when D = 26. In later chapters we will obtain a deeper understanding of how this comes about and what it implies. We conclude this section with a few more comments about spin. The light-cone quantization singles out two directions and leaves manifest only the S O(D − 2) that acts in the transverse directions. The spin generators for the transverse directions are S = −i ij

∞ 1

n n=1

(αi−n αjn − αj−n αin ) ;

(1.3.41)

antisymmetry on ij together with Lorentz invariance allows no zero-point constant. For a massless particle, the full S O(D −2) spin symmetry is made manifest by choosing the momentum to lie along the 1-direction singled out in the quantization. For a massive particle, only an S O(D − 2) subgroup of the S O(D − 1) spin symmetry will be manifest in light-cone quantization, but this is still quite useful. For example, the (D−1)-dimensional vector representation of S O(D − 1) breaks up into an invariant and a (D−2)-vector under the S O(D − 2) acting on the transverse directions, v = (v 1 , 0, . . . , 0) + (0, v 2 , . . . , v D−1 ) .

(1.3.42)

Thus, if a massive particle is in the vector representation of S O(D − 1), we will see a scalar and a vector when we look at the transformation properties under S O(D − 2). This idea extends to any representation: one can always reconstruct the full S O(D − 1) spin representation from the behavior under S O(D − 2). The higher excited states of the string, which are massive, do form full representations of S O(D − 1). One can check this by hand for a few levels, and in a more complete analysis it follows from the existence of the full set of Lorentz generators. We will show this indirectly in the coming chapters, by ﬁnding a consistent covariant quantization. At level N, the maximum eigenvalue of a given spin component, say S 23 , is N, obtained by acting N times with α2−1 + iα3−1 . Thus, S 23 ≤ 1 + α m2 .

(1.3.43)

The slope in this inequality is known as the Regge slope. Meson resonances obey a linear relation of this form, with α ∼ (1 GeV)−2 . For this and other reasons, string theory was originally proposed in 1970 as a theory of the strong interaction. Within a few years, however, the S U(3) gauge theory QCD was discovered to be the correct theory, with string-like behavior only in certain limits.

1.4 Closed and unoriented strings

25

Now that we are considering strings as the uniﬁed theory of particle physics, gravity, and quantum mechanics, α will be of the order of the natural scale determined by the fundamental constants of gravity and quantum mechanics, MP−2 . In particular, masses that are not zero in string theory are of order MP . This is so large compared to experimentally accessible energy scales that these particles appear only in virtual states. Thus, we will be especially concerned with the massless string spectrum, since this must include all the particles of the Standard Model. Of course, most known particles are massive, but these masses are so small compared to MP that they are zero to ﬁrst approximation and become non-zero due to small symmetry-breaking eﬀects. Some of the interest in string theory has been driven by the possibility that the strong interaction theory QCD is equivalent to a string theory, at least in some approximation. There has been cross-fertilization between this idea and the idea of strings as a fundamental theory. The connection between QCD and string theory is outside the focus of this book, but the recent developments described in chapter 14 have also given new insight in this direction. 1.4

Closed and unoriented strings

The light-cone quantization of the closed string is quite similar to that of the open string. Again impose the gauge conditions (1.3.8). In the open string, these determined the gauge completely. In the closed string there is still some extra coordinate freedom, σ = σ + s(τ) mod ,

(1.4.1)

because the point σ = 0 can be chosen anywhere along the string. Most of this remaining freedom can be ﬁxed by the additional gauge condition γτσ (τ, 0) = 0 .

(1.4.2)

That is, the line σ = 0 is orthogonal to the lines of constant τ. This determines the line σ = 0 except for an overall σ-translation. Conditions (1.3.8) and (1.4.2) thus ﬁx all of the gauge freedom except for τ-independent translations of σ, σ = σ + s mod .

(1.4.3)

We will deal with this extra gauge freedom later. The analysis is now parallel to that for the open string. The Lagrangian, canonical momenta, Hamiltonian, and equation of motion are as in the open string, eqs. (1.3.16)–(1.3.21). The general periodic solution to the

26

1 A ﬁrst look at strings

equation of motion is

α 1/2 pi τ + i p+ 2 ∞ i α 2πin(σ − cτ) 2πin(σ + cτ) α˜in n × + . exp − exp n n n=−∞

X i (τ, σ) = xi +

n=0

(1.4.4) In the closed string there are two independent sets of oscillators, αin and ˜αin , corresponding to left-moving and right-moving waves along the string. In the open string the boundary condition at the endpoints tied these together. The independent degrees of freedom are again the transverse oscillators and the transverse and longitudinal center-of-mass variables, αin , α˜in , xi , pi , x− , p+ ,

(1.4.5)

with canonical commutators

kµ

[x− , p+ ] = −i , [xi , pj ] = iδ ij , [αim , αjn ] = mδ ij δm,−n ,

(1.4.6a) (1.4.6b) (1.4.6c)

[˜αim , ˜αjn ] = mδ ij δm,−n .

(1.4.6d)

Starting from the state |0, 0; k , which has center-of-mass momentum and is annihilated by αim s and ˜αim s for m > 0, the general state is ˜ k = |N, N;

D−1 ∞

i=2 n=1

˜

(αi−n )Nin (˜αi−n )Nin |0, 0; k . ˜ in !)1/2 (nNin Nin !nN˜ in N

(1.4.7)

The mass formula is m2 = 2p+ H − pi pi

∞ 2 i i i i ˜ = (α α + ˜α−n ˜αn ) + A + A α n=1 −n n 2 ˜ + A + A) ˜ . (N + N (1.4.8) α We have broken up the level and zero-point constant into the part from the right-moving modes and the part from the left. Summing zero-point energies again gives =

2−D . (1.4.9) 24 There is one further restriction on the state due to the remaining gauge freedom, the σ-translations (1.4.3). The physical spectrum is obtained ˜= A=A

27

1.4 Closed and unoriented strings

by restricting to gauge-invariant states. The operator that generates the σ-translations is P =−

0

dσ Πi ∂σ X i

∞ 2π ˜ =− (αi αi − ˜αi−n ˜αin ) + A − A n=1 −n n

2π ˜ . (N − N) States must therefore satisfy =−

(1.4.10)

˜ . N=N

(1.4.11)

The lightest closed string state is 2−D , 6α which is again a tachyon. The ﬁrst excited states are |0, 0; k ,

m2 =

(1.4.12)

26 − D . (1.4.13) 6α As with the open string, these states do not add up to complete representations of S O(D − 1) and so this level must be massless. Thus, ˜ = −1 , D = 26 . A=A (1.4.14) αi−1 ˜αj−1 |0, 0; k ,

m2 =

The states (1.4.13) transform as a 2-tensor under S O(D − 2). This is a reducible representation: it decomposes into a symmetric traceless tensor, an antisymmetric tensor, and a scalar. That is, any tensor eij can be decomposed

1 ij 2 1 ij 1 δ ij ekk + δ ij ekk , (1.4.15) e + eji − e − eji + e = 2 D−2 2 D−2 and the three separate terms do not mix under rotations. ˜ in are independent except for the N = N ˜ At any mass level, Nin and N 2 constraint. Thus, the closed string spectrum at m = 4(N − 1)/α is the product of two copies of the m2 = (N − 1)/α level of the open string. Let us now discuss some general issues. The two-dimensional diﬀ invariance removed two families of normal modes from the spectrum. If we tried to make a covariant theory without this invariance, we would have to generalize the transverse commutator (1.3.25) to ij

[αµm , ανn ] = mη µν δm,−n .

(1.4.16)

Lorentz invariance forces the timelike oscillators to have the wrong-sign commutator. A state with an odd number of timelike excitations will then have a negative norm. This is inconsistent with quantum mechanics,

28

1 A ﬁrst look at strings

since the norm is a probability. In fact a theory based on (1.4.16) came before the string theory we have described, and the latter was discovered by requiring that the negative norm states never be produced in physical processes. The commutator (1.4.16) arises in a covariant quantization of string theory (which we shall use beginning in the next chapter), with the coordinate invariance appearing through constraints on the spectrum to eliminate the unphysical states. The existence of a massless vector in the open string theory and of the massless symmetric and antisymmetric tensors in the closed string theory is striking. General principles require that a massless vector couple to a conserved current and therefore that the theory have a gauge invariance. This is our ﬁrst example of the massless gauge particles that are present in all fundamental string theories. Actually, this particular gauge boson, call it the photon, is not so interesting, because the gauge group is only U(1) (there is only one photon) and because it turns out that all particles in the theory are neutral. In chapter 6 we will discuss a simple generalization of open string theory, adding Chan–Paton degrees of freedom at the string endpoints, which leads to U(n), S O(n), and S p(n) gauge groups. Similarly, a massless symmetric tensor particle must couple to a conserved symmetric tensor source. The only such source is the energy-momentum tensor; some additional possibilities arise in special cases, but these are not relevant here. Coupling to this in a consistent way requires the theory to have spacetime coordinate invariance. Thus the massless symmetric tensor is the graviton, and general relativity is contained as one small piece of the closed string theory. The massless antisymmetric tensor is known as a 2-form gauge boson. As we will see in section 3.7, there is also a local spacetime symmetry associated with it. These spacetime gauge and coordinate invariances were not evident in our starting point, the Nambu–Goto or Polyakov theory. We have certainly discovered them in a backwards way: adding up zero-point energies, requiring a Lorentz-invariant spectrum, and citing general results about interactions of massless particles. In gauge theory and in general relativity, one starts with a spacetime symmetry principle. In string theory, too, it would seem that we should ﬁrst ﬁgure out what the full spacetime symmetry is, and use this to deﬁne the theory. There are various attempts in this direction, but a complete picture has not yet emerged. For now, let us note one thing: we have observed that the local world-sheet symmetry diﬀ removes the unphysical timelike and longitudinal normal modes which would lead to negative norm string states. For example, the open string photon would otherwise have D states rather than D−2. From the spacetime point of view, it is the local spacetime invariances that remove these same unphysical states. So there is at least some connection between the local symmetries of the world-sheet and those of spacetime.

1.4 Closed and unoriented strings

29

Anticipating our interest in four dimensions, let us look at the massless string states obtained by exciting only oscillators in the 2- and 3-directions, and classifying them by the helicity λ = S 23 . One ﬁnds that the photon has λ = ±1 and the graviton λ = ±2, justifying the names. The closed string scalar and antisymmetric tensor each give rise to a λ = 0 state. These are called the dilaton and the axion, respectively, and we will discuss their physics at various points. We conclude this section with a discussion of one other generalization, the unoriented string theory. The theories we have been discussing are oriented string theories. We have not considered the coordinate transformation σ = − σ ,

τ = τ ,

(1.4.17)

which changes the orientation (handedness) of the world-sheet. This symmetry is generated by the world-sheet parity operator Ω. Carrying out (1.4.17) twice gives the identity, so Ω2 = 1 and the eigenvalues of Ω are ±1. From the mode expansions (1.3.22) and (1.4.4) we see that Ωαin Ω−1 = (−1)n αin

(1.4.18)

Ωαin Ω−1 = α˜in , Ω˜αin Ω−1 = αin

(1.4.19a) (1.4.19b)

in the open string and

in the closed string. We deﬁne the phase of Ω by ﬁxing Ω = +1 for the ground states |0; k and |0, 0; k ; later we will see that this choice is required in order for Ω to be conserved by interactions. Then Ω|N; k = (−1)N |N; k , ˜ k = |N, ˜ N; k . Ω|N, N;

(1.4.20a) (1.4.20b)

There are consistent interacting string theories, the unoriented theories, in which only the Ω = +1 states are kept. Focusing on the massless states, the open string photon has Ω = −1 and is absent in the unoriented theory. Acting on the massless tensor states in the closed string, the parity operator Ω takes eij to eji , so the graviton and dilaton are present in the unoriented theory while the antisymmetric tensor is not. Notice that both the open and closed string tachyons survive in the unoriented theory; we will have to work harder to remove these. Let us mention two other constraints that will emerge from the study of the interactions. First, it is possible to have a consistent theory with only closed strings, or with closed and open strings, but not with open strings alone: closed strings can always be produced in the scattering of open strings. Second, oriented or unoriented open strings can only couple to

30

1 A ﬁrst look at strings

closed strings of the same type. We list the possible combinations together with their massless spectra, with Gµν representing the graviton, Bµν the antisymmetric tensor, Φ the dilaton, and Aµ the photon: 1. Closed oriented bosonic string: Gµν , Bµν , Φ . 2. Closed unoriented bosonic string: Gµν , Φ . 3. Closed plus open oriented bosonic string: Gµν , Bµν , Φ, Aµ . 4. Closed plus open unoriented bosonic string: Gµν , Φ . All of these have the graviton and dilaton, as well as the tachyon. In chapter 6 we will discuss more general open string theories, with Chan– Paton degrees of freedom at the endpoints. The massless oriented open strings will then be U(n) gauge bosons, and the massless unoriented open strings will be S O(n) or S p(n) gauge bosons. Exercises 1.1 (a) Show that in the nonrelativistic limit the action Spp has the usual nonrelativistic form, kinetic energy minus potential energy, with the potential energy being the rest mass. (b) Show that for a string moving nonrelativistically, the Nambu–Goto action reduces to a kinetic term minus a potential term proportional to the length of the string. Show that the kinetic energy comes only from the transverse velocity of the string. Calculate the mass per unit length, as determined from the potential term and also from the kinetic term. 1.2 Show that classical equations of motion imply that the ends of the open string move at the speed of light. 1.3 For world-sheets with boundary, show that

1 1 dτ dσ (−γ)1/2 R + ds k 4π M 2π ∂M is Weyl-invariant. Here ds is the proper time along the boundary in the metric γab , and k is the geodesic curvature of the boundary, χ=

k = ±ta nb ∇a tb , where ta is a unit vector tangent to the boundary and na is an outward pointing unit vector orthogonal to ta . The upper sign is for a Lorentzian world-sheet and the lower sign for a Euclidean world-sheet. 1.4 Show that the open string states at levels m2 = 1/α and 2/α form complete representations of S O(D − 1). You will need to work out the S O(D − 2) content of various symmetric and antisymmetric tensor repre-

Exercises

31

sentations of S O(D − 1). You can carry this to higher levels if you like, but you will need tensors of mixed symmetry and Young tableaux. 1.5 Extend the sum (1.3.32) to the ‘twisted’ case ∞

(n − θ)

n=1

with θ a constant. That is, kσ = (n − θ)π/. The answer is given in eq. (2.9.19). You should ﬁnd that the cutoﬀ-dependent term is independent of θ. In the following exercises one has the usual open or closed string boundary conditions (Neumann or periodic) on X µ for µ = 0, . . . , 24 but a diﬀerent boundary condition on X 25 . Each of these has an important physical interpretation, and will be developed in detail in chapter 8. Find the mode expansion, the mass spectrum, and (for the closed string) the constraint from σ-translation invariance in terms of the occupation numbers. In some cases you need the result of exercise 1.5. 1.6 Open strings with X 25 (τ, 0) = 0 ,

X 25 (τ, ) = y

with y a constant. This is an open string with both ends on D-branes. 1.7 Open strings with X 25 (τ, 0) = 0 ,

∂σ X 25 (τ, ) = 0 .

This is an open string with one end on a D-brane and one end free. 1.8 Closed strings with X 25 (τ, σ + ) = X 25 (τ, σ) + 2πR with R a constant. This is a winding string in toroidal (periodic) compactiﬁcation. In this case p25 must be a multiple of 1/R. 1.9 Closed strings with X 25 (τ, σ + ) = −X 25 (τ, σ) . This is a twisted string in orbifold compactiﬁcation.

2 Conformal ﬁeld theory

In this chapter we develop a number of necessary ideas and techniques from the world-sheet quantum ﬁeld theory, including the operator product expansion, conformal invariance, the Virasoro algebra, and vertex operators. The focus is on conformally invariant ﬁeld theory in two ﬂat dimensions; as we will see in the next chapter, this is what we are left with after ﬁxing the local symmetries of the string world-sheet. 2.1

Massless scalars in two dimensions

We will start with the example of D free scalar ﬁelds in two dimensions, X µ (σ 1 , σ 2 ). We will refer to these two dimensions as the world-sheet, anticipating the application to string theory. The action is 1 S= 4πα

d2 σ ∂1 X µ ∂1 X µ + ∂2 X µ ∂2 X µ .

(2.1.1)

This is the Polyakov action (1.2.13), except that the world-sheet metric γab has been replaced with a ﬂat Euclidean metric δab , signature (+, +). The overall sign change of the action is a result of the Euclidean convention (A.1.31). As we will see, most string calculations are carried out on a Euclidean world-sheet. At least for ﬂat metrics, the relation between Euclidean and Minkowski amplitudes is given by a standard analytic continuation, explained in the appendix. In fact, the results of the present chapter apply equally to a Minkowski world-sheet, and in the ﬁrst seven sections all equations make sense if σ 2 is replaced with iσ 0 . For the index µ we still take the ﬂat Minkowski metric. It is straightforward to quantize the action (2.1.1) canonically, ﬁnding the spectrum, vacuum expectation values, and so on. We have done essentially this in chapter 1, after having gone to light-cone gauge. Here we will take a somewhat diﬀerent route, developing ﬁrst various local properties such 32

2.1 Massless scalars in two dimensions

33

as equations of motion, operator products, Ward identities, and conformal invariance, before working our way around to the spectrum. It will be eﬃcient for us to use the path integral formalism. This is reviewed in the appendix. We will be using the path integral representation primarily to derive operator equations (to be deﬁned below); these can also be derived in a Hilbert space formalism. It is very useful to adopt complex coordinates z = σ 1 + iσ 2 ,

¯z = σ 1 − iσ 2 .

(2.1.2)

We will use a bar for the complex conjugates of z and other simple variables, and a star for the complex conjugates of longer expressions. Deﬁne also 1 ∂z = (∂1 − i∂2 ) , 2

1 ∂¯z = (∂1 + i∂2 ) . 2

(2.1.3)

These derivatives have the properties ∂z z = 1 ,

∂z ¯z = 0 ,

∂¯z z = 0 ,

∂¯z ¯z = 1 .

(2.1.4)

It is conventional to abbreviate ∂z to ∂ and ∂¯z to ∂¯ when this is not ambiguous. For a general vector v a , deﬁne in the same way v z = v 1 +iv 2 ,

1 vz = (v 1 −iv 2 ) , 2

v¯z = v 1 −iv 2 ,

1 v¯z = (v 1 +iv 2 ) . (2.1.5) 2

For the indices 1, 2 the metric is the identity and we do not distinguish between upper and lower, while the complex indices are raised and lowered with gz¯z = g¯z z =

1 , 2

gzz = g¯z¯z = 0 ,

g z¯z = g¯z z = 2 ,

g zz = g¯z¯z = 0 . (2.1.6)

Note also that d2 z = 2dσ 1 dσ 2

(2.1.7)

with the factor of 2 from the Jacobian,1 and that d2 z | det g|1/2 = dσ 1 dσ 2 . We deﬁne

so that δ 2 (z, ¯z ) =

1

d2 z δ 2 (z, ¯z ) = 1

1 1 2 2 δ(σ )δ(σ ).

(2.1.8)

Another useful result is the divergence

This diﬀers from much of the literature, where d2 z is deﬁned as dσ 1 dσ 2 . Correlated with this, the 1 2 z ). 2 is omitted from the deﬁnition of δ (z, ¯

34

2 Conformal ﬁeld theory

theorem in complex coordinates,

R

d2 z (∂z v z + ∂¯z v¯z ) = i

∂R

(v z d¯z − v¯z dz) ,

(2.1.9)

where the contour integral circles the region R counterclockwise. In this notation the action is S=

1 2πα

¯ µ d2 z ∂X µ ∂X

(2.1.10)

and the classical equation of motion is ¯ µ (z, ¯z ) = 0 . ∂∂X

(2.1.11)

The notation X µ (z, ¯z ) may seem redundant, since the value of z determines the value of ¯z , but it is useful to reserve the notation f(z) for ﬁelds whose equation of motion makes them analytic (equivalently holomorphic) functions of z. Writing the equation of motion as µ ¯ ¯ µ ) = ∂(∂X )=0, ∂(∂X

(2.1.12)

¯ µ is antiholomorphic it follows that ∂X µ is holomorphic and that ∂X ¯ µ (¯z ). (holomorphic in ¯z ), hence the notations ∂X µ (z) and ∂X 2 0 Under the Minkowski continuation σ = iσ , a holomorphic ﬁeld becomes a function only of σ 0 − σ 1 and an antiholomorphic ﬁeld a function only of σ 0 + σ 1 . We thus use as synonyms holomorphic = left-moving , antiholomorphic = right-moving .

(2.1.13a) (2.1.13b)

This terminology is chosen to have maximal agreement with the literature, though for it to hold literally we would need to draw σ 1 increasing from right to left. We will tend to use the Euclidean terms early on and shift to the Minkowski terms as we discuss more of the spacetime physics. Expectation values are deﬁned by the path integral,

F[X] =

[dX] exp(−S )F[X] ,

(2.1.14)

where F[X] is any functional of X, such as a product of local operators. The path integral over X 0 is a wrong-sign Gaussian, so it should be understood to be deﬁned by the analytic continuation X 0 → −iX D . The reader should not be distracted by this; we will discuss it further in the next chapter. We do not normalize F[X] by dividing by 1 . The path integral of a total derivative is zero. This is true for ordinary bosonic path integrals, which can be regarded as the limit of an inﬁnite number of ordinary integrals, as well as for more formal path integrals as

35

2.1 Massless scalars in two dimensions with Grassmann variables. Then

δ exp(−S ) δXµ (z, ¯z ) δS = − [dX] exp(−S ) δXµ (z, ¯z )

0=

[dX]

=− =

δS δXµ (z, ¯z )

1 ¯ µ ¯ ∂ ∂X (z, z ) . πα

(2.1.15)

The same calculation goes through if we have arbitrary additional insertions ‘. . .’ in the path integral, as long as none of these additional insertions is at z. Thus

¯ µ (z, ¯z ) . . . ∂∂X

=0.

(2.1.16)

We can regard the additional insertions as preparing arbitrary initial and ﬁnal states (or we could do the same thing with boundary conditions). The path integral statement (2.1.16) is thus the same as the statement in the Hilbert space formalism that ˆ µ (z, ¯z ) = 0 ∂∂¯X

(2.1.17)

ˆ µ (z, ¯z ). Thus we refer holds for all matrix elements of the operator X to relations that hold in the sense (2.1.16) as operator equations. The statement (2.1.17) is Ehrenfest’s theorem that the classical equations of motion translate into operator equations. The notation ‘. . .’ in the path integral (2.1.16) implicitly stands for insertions that are located away from z, but it is interesting to consider also the case in which there is an insertion that might be coincident with z:

[dX]

0=

=

δ exp(−S )X ν (z , ¯z ) δXµ (z, ¯z )

[dX] exp(−S ) η µν δ 2 (z − z , ¯z − ¯z ) +

= η µν δ 2 (z − z , ¯z − ¯z ) +

1 µ ν ¯ ¯ ∂ ∂ X (z, z )X (z , z ) ¯ z z πα

1 ∂z ∂¯z X µ (z, ¯z )X ν (z , ¯z ) . πα

(2.1.18)

That is, the equation of motion holds except at coincident points. Again this goes through with arbitrary additional insertions ‘. . .’ in the path integral, as long as none of these additional ﬁelds is at (z, ¯z ) or (z , ¯z ): µ 1 ν µν 2 ¯ ¯ ¯ ¯ ∂ ∂ X (z, z )X (z , z ) . . . = −η δ (z − z , z − z ) . . . . (2.1.19) z ¯z πα

36

2 Conformal ﬁeld theory

Thus, 1 ∂z ∂¯z X µ (z, ¯z )X ν (z , ¯z ) = −η µν δ 2 (z − z , ¯z − ¯z ) (2.1.20) πα holds as an operator equation. In the Hilbert space formalism, the product in the path integral becomes a time-ordered product, and the delta function comes from the derivatives acting on the time-ordering. This connection is developed further in the appendix. In free ﬁeld theory, it is useful to introduce the operation of normal ordering. Normal ordered operators, denoted : A :, are deﬁned as follows, : X µ (z, ¯z ) : = X µ (z, ¯z ) , : X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) : = X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) +

(2.1.21a) α 2

η µν ln |z12 |2 , (2.1.21b)

where zij = zi − zj .

(2.1.22)

The reader may be familiar with normal ordering deﬁned in terms of raising and lowering operators; these two deﬁnitions will be related later. The point of this deﬁnition is the property ∂1 ∂¯1 : X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) : = 0 .

(2.1.23)

This follows from the operator equation (2.1.20) and the diﬀerential equation ∂∂¯ ln |z|2 = 2πδ 2 (z, ¯z ) .

(2.1.24)

Eq. (2.1.24) is obvious for z = 0 because ln |z|2 = ln z + ln ¯z , and the normalization of the delta function is easily checked by integrating both sides using eq. (2.1.9). 2.2

The operator product expansion

The basic object of interest in string perturbation theory will be the path integral expectation value of a product of local operators,

Ai1 (z1 , ¯z1 ) Ai2 (z2 , ¯z2 ) . . . Ain (zn , ¯zn ) ,

(2.2.1)

where Ai is some basis for the set of local operators. It is particularly important to understand the behavior of this expectation value in the limit that two of the operators are taken to approach one another. The tool that gives a systematic description of this limit is the operator product expansion (OPE), illustrated in ﬁgure 2.1. This states that a product of two

37

2.2 The operator product expansion

z3 z1 z2 z4

Fig. 2.1. Expectation value of a product of four local operators. The OPE gives the asymptotics as z1 → z2 as a series where the pair of operators at z1 and z2 is replaced by a single operator at z2 . The radius of convergence is the distance to the nearest other operator, indicated by the dashed circle.

local operators close together can be approximated to arbitrary accuracy by a sum of local operators, Ai (σ1 )Aj (σ2 ) =

ckij (σ1 − σ2 )Ak (σ2 ) .

(2.2.2)

k

Again this is an operator statement, meaning that it holds inside a general expectation value

Ai (σ1 )Aj (σ2 ) . . . =

ckij (σ1 − σ2 ) Ak (σ2 ) . . .

(2.2.3)

k

as long as the separation between σ1 and σ2 is small compared to the distance to any other operator. The coeﬃcient functions ckij (σ1 − σ2 ), which govern the dependence on the separation, depend on i, j, and k but not on the other operators in the expectation value; the dependence on the latter comes only through the expectation value on the right-hand side of eq. (2.2.3). The terms are conventionally arranged in order of decreasing size in the limit as σ1 → σ2 . This is analogous to an ordinary Taylor series, except that the coeﬃcient functions need not be simple powers and can in fact be singular as σ1 → σ2 , as we will see even in the simplest example. Just as the Taylor series plays a central role in calculus, the OPE plays a central role in quantum ﬁeld theory. We will give now a derivation of the OPE for the X µ theory, using the special properties of free ﬁeld theory. In section 2.9 we will give a derivation for any conformally invariant ﬁeld theory. We have seen that the normal ordered product satisﬁes the naive equation of motion. Eq. (2.1.23) states that the operator product is a harmonic function of (z1 , ¯z1 ). A simple result from the theory of complex variables is that a harmonic function is locally the sum of a holomorphic and an

38

2 Conformal ﬁeld theory

antiholomorphic function. In particular, this means that it is nonsingular as z1 → z2 and can be freely Taylor expanded in z12 and ¯z12 . Thus, α X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) = − η µν ln |z12 |2 + : X ν X µ (z2 , ¯z2 ) : 2 ∞ 1

+ (z12 )k : X ν ∂k X µ (z2 , ¯z2 ) : + (¯z12 )k : X ν ∂¯k X µ (z2 , ¯z2 ) : . k! k=1 (2.2.4) Terms with mixed ∂∂¯ derivatives vanish by the equation of motion. This equation and many others simplify in units in which α = 2, which is the most common convention in the literature. However, several other conventions are also used, so it is useful to keep α explicit. For example, in open string theory equations simplify when α = 12 . Eq. (2.2.4) has the form of an OPE. Like the equation of motion (2.1.23) from which it was derived, it is an operator statement. For an arbitrary expectation value involving the product X µ (z1 , ¯z1 )X ν (z2 , ¯z2 ) times ﬁelds at other points, it gives the behavior for z1 → z2 as an inﬁnite series, each term being a known function of z12 and/or ¯z12 times the expectation value with a local operator replacing the pair. OPEs are usually used as asymptotic expansions, the ﬁrst few terms giving the dominant behavior at small separation. Most of our applications will be of this type, and we will often write OPEs as explicit singular terms plus unspeciﬁed nonsingular remainders. The use of ‘∼’ in place of ‘=’ will mean ‘equal up to nonsingular terms.’ In fact, OPEs are actually convergent in conformally invariant ﬁeld theories. This will be very important to us in certain applications: it makes it possible to reconstruct the entire theory from the coeﬃcient functions. As an example, the freeﬁeld OPE (2.2.4) has a radius of convergence in any given expectation value which is equal to the distance to the nearest other insertion in the path integral. The operator product is harmonic except at the positions of operators, and in particular inside the dashed circle of ﬁgure 2.1, and convergence can then be shown by a standard argument from the theory of complex variables. The various operators on the right-hand side of the OPE (2.2.4) involve products of ﬁelds at the same point. Usually in quantum ﬁeld theory such a product is divergent and must be appropriately cut oﬀ and renormalized, but here the normal ordering renders it well-deﬁned. Normal ordering is a convenient way to deﬁne composite operators in free ﬁeld theory. It is of little use in most interacting ﬁeld theories, because these have additional divergences from interaction vertices approaching the composite operator or one another. However, many of the ﬁeld theories that we will be

39

2.2 The operator product expansion

interested in are free, and many others can be related to free ﬁeld theories, so it will be worthwhile to develop normal ordering somewhat further. The deﬁnition of normal ordering for arbitrary numbers of ﬁelds can be given recursively as : X µ1 (z1 , ¯z1 ) . . . X µn (zn , ¯zn ) : = X µ1 (z1 , ¯z1 ) . . . X µn (zn , ¯zn ) + subtractions ,

(2.2.5)

where the sum runs over all ways of choosing one, two, or more pairs of ﬁelds from the product and replacing each pair with 12 α η µi µj ln |zij |2 . For example, : X µ1 (z1 , ¯z1 )X µ2 (z2 , ¯z2 )X µ3 (z3 , ¯z3 ) : = X µ1 (z1 , ¯z1 )X µ2 (z2 , ¯z2 )X µ3 (z3 , ¯z3 ) α µ1 µ2 2 µ3 ln |z12 | X (z3 , ¯z3 ) + 2 permutations . (2.2.6) + η 2 We leave it to the reader to show that the deﬁnition (2.2.5) retains the desired property that the normal ordered product satisﬁes the naive equation of motion. The deﬁnition can be compactly summarized as α

: F := exp

4

d2 z1 d2 z2 ln |z12 |2

δ δ F, µ δX (z1 , ¯z1 ) δXµ (z2 , ¯z2 )

(2.2.7)

where F is any functional of X. This is equivalent to eq. (2.2.5): the double derivative in the exponent contracts each pair of ﬁelds, and the exponential sums over any number of pairs with the factorial canceling the number of ways the derivatives can act. As an example of the use of this formal expression, act on both sides with the inverse exponential to obtain δ α δ d2 z1 d2 z2 ln |z12 |2 :F: F = exp − 4 δX µ (z1 , ¯z1 ) δXµ (z2 , ¯z2 ) = :F: +

contractions ,

(2.2.8)

where a contraction is the opposite of a subtraction: sum over all ways of choosing one, two, or more pairs of ﬁelds from : F : and replacing each pair with − 12 α η µi µj ln |zij |2 . The OPE for any pair of operators can be generated from : F : : G : = : FG : +

cross-contractions

(2.2.9)

for arbitrary functionals F and G of X. The sum now runs over all ways of contracting pairs with one ﬁeld in F and one in G. This can also be written δ δ α 2 2 2 d z1 d z2 ln |z12 | : FG : , : F : : G : = exp − 2 δXFµ (z1 , ¯z1 ) δXGµ (z2 , ¯z2 ) (2.2.10)

40

2 Conformal ﬁeld theory

where the functional derivatives act only on the ﬁelds in F or G respectively. This follows readily from eq. (2.2.7). As an example, : ∂X µ (z)∂Xµ (z) : : ∂ X ν (z )∂ Xν (z ) : = : ∂X µ (z)∂Xµ (z)∂ X ν (z )∂ Xν (z ) : α − 4 · (∂∂ ln |z − z |2 ) : ∂X µ (z)∂ Xµ (z ) : 2 2 α + 2 · η µµ − ∂∂ ln |z − z |2 2 2 Dα 2α ∼ − : ∂ X µ (z )∂ Xµ (z ) : 4 2(z − z ) (z − z )2 2α − : ∂2 X µ (z )∂ Xµ (z ) : . (2.2.11) z − z The second term in the equality comes from the four ways of forming a single pair and the third from the two ways of forming two pairs. In the ﬁnal line we have put the OPE in standard form by Taylor expanding inside the normal ordering to express everything in terms of local operators at z and putting the most singular terms ﬁrst. Another important example is F = eik1 ·X(z,¯z ) ,

G = eik2 ·X(0,0) .

(2.2.12)

The variations δ/δXFµ and δ/δXGµ give factors of ik1µ and ik2µ respectively, so the general result (2.2.10) becomes α

: eik1 ·X(z,¯z ) : : eik2 ·X(0,0) : = exp = |z|

2

α k1 ·k2

k1 · k2 ln |z|2 : eik1 ·X(z,¯z ) eik2 ·X(0,0) : : eik1 ·X(z,¯z ) eik2 ·X(0,0) : .

(2.2.13)

To derive the OPE, Taylor expand inside the normal ordering to give

: eik1 ·X(z,¯z ) : : eik2 ·X(0,0) : = |z|α k1 ·k2 : ei(k1 +k2 )·X(0,0) [1 + O(z, ¯z )] : .

(2.2.14)

The exercises give further practice with normal ordering and the free-ﬁeld OPE. Note that the OPEs (2.2.2), (2.2.4), and so on have been written asymmetrically in σ1 and σ2 , expanding around the latter point. They can also be cast in symmetric form by Taylor expanding the right-hand sides around (σ1 + σ2 )/2. The coeﬃcient functions for the symmetric form behave simply under interchange of the two operators, ckij (σ1 − σ2 )sym = ±ckji (σ2 − σ1 )sym ,

(2.2.15)

where the minus sign appears if Ai and Aj are both anticommuting. The asymmetric form is usually more convenient for calculation, so when the

41

2.3 Ward identities and Noether’s theorem

symmetry properties of the coeﬃcient functions are needed one can work them out in the symmetric form and then convert to the asymmetric form. 2.3

Ward identities and Noether’s theorem

World-sheet symmetries of course play an important role in string theory. In this section we ﬁrst derive some general consequences of symmetry in ﬁeld theory. Consider a general ﬁeld theory with action S [φ] in d spacetime dimensions, with φα (σ) denoting general ﬁelds. Let there be a symmetry φα (σ) = φα (σ) + δφα (σ) ,

(2.3.1)

where δφα is proportional to an inﬁnitesimal parameter . The product of the path integral measure and the weight exp(−S ) is invariant, [dφ ] exp(−S [φ ]) = [dφ] exp(−S [φ]) .

(2.3.2)

A continuous symmetry in ﬁeld theory implies the existence of a conserved current (Noether’s theorem) and also Ward identities, which constrain the operator products of the current. To derive these results consider the change of variables, φα (σ) = φα (σ) + ρ(σ)δφα (σ) .

(2.3.3)

This is not a symmetry, the transformation law being altered by the inclusion of an arbitrary function ρ(σ). The path integral measure times exp(−S ) would be invariant if ρ were a constant, so its variation must be proportional to the gradient ∂a ρ, [dφ ] exp(−S [φ ])

= [dφ] exp(−S [φ]) 1 +

i 2π

dd σ g 1/2 j a (σ)∂a ρ(σ) + O(2 ) . (2.3.4)

The unknown coeﬃcient j a (σ) comes from the variation of the measure and the action, both of which are local, and so it must be a local function of the ﬁelds and their derivatives. Take the function ρ to be nonzero only in a small region, and consider a path integral with general insertions ‘. . .’ outside this region; the insertions are therefore invariant under (2.3.3). Invariance of the path integral under change of variables gives

0= =

[dφ ] exp(−S [φ ]) . . . −

2πi

[dφ] exp(−S [φ]) . . .

dd σ g 1/2 ρ(σ) ∇a j a (σ) . . . ,

(2.3.5)

where the limited support of ρ has allowed us to integrate by parts. Thus

42

2 Conformal ﬁeld theory

we have ∇a j a = 0

(2.3.6)

as an operator equation. This is Noether’s theorem, which is developed further in exercise 2.5. To derive the Ward identity, let ρ(σ) be 1 in some region R and 0 outside R. Also, include in the path integral some general local operator A(σ0 ) at a point σ0 inside R, and the usual arbitrary insertions ‘. . .’ outside. Proceeding as above we obtain the operator relation δA(σ0 ) + dd σ g 1/2 ∇a j a (σ)A(σ0 ) = 0 . (2.3.7) 2πi R Equivalently, ∇a j a (σ)A(σ0 ) = g −1/2 δ d (σ − σ0 )

2π δA(σ0 ) + total σ-derivative . (2.3.8) i

The divergence theorem gives

2π δA(σ0 ) (2.3.9) i ∂R with dA the area element and na the outward normal. This relates the integral of the current around the operator to the variation of the operator. Going to two ﬂat dimensions this becomes

dA na j a A(σ0 ) =

2π (2.3.10) δA(z0 , ¯z0 ) . ∂R Again we drop indices, j ≡ jz , ˜ ≡ j¯z ; notice that on a current the omitted indices are implicitly lower. We use a tilde rather than a bar on ˜ because this is not the adjoint of j. The Minkowski density j0 is in general Hermitean, so the Euclidean j2 with an extra factor of i is anti-Hermitean, and (jz )† = 12 (j1 − ij2 )† = jz . It is important that Noether’s theorem and the Ward identity are local properties, which do not depend on whatever boundary conditions we might have far away, nor even on whether these are invariant under the symmetry. In particular, since the function ρ(σ) is nonzero only inside R, the symmetry transformation need only be deﬁned there. In conformally invariant theories it is usually the case that jz is holomorphic and j¯z antiholomorphic (except for singularities at the other ﬁelds). In this case the currents (jz , 0) and (0, j¯z ) are separately conserved. The integral (2.3.10) then picks out the residues in the OPE, (jdz − ˜ d¯z )A(z0 , ¯z0 ) =

1 δA(z0 , ¯z0 ) . (2.3.11) i Here ‘Res’ and ‘Res’ are the coeﬃcients of (z − z0 )−1 and (¯z − ¯z0 )−1 respectively. This form of the Ward identity is particularly convenient. Resz→z0 j(z)A(z0 , ¯z0 ) + Res¯z →¯z0 ˜ (¯z )A(z0 , ¯z0 ) =

2.4 Conformal invariance

43

The world-sheet current was deﬁned with an extra factor of 2πi relative to the usual deﬁnition in ﬁeld theory in order to make this OPE simple. As an example, return to the free massless scalar and consider the spacetime translation δX µ = aµ . Under δX µ (σ) = ρ(σ)aµ , aµ δS = d2 σ ∂ a X µ ∂a ρ . (2.3.12) 2πα This is of the claimed form (2.3.5) with Noether current aµ jaµ , where i ∂a X µ . (2.3.13) α The components are holomorphic and antiholomorphic as expected. For the OPE of this current with the exponential operator one ﬁnds jaµ =

k µ ik·X(0,0) :e :, (2.3.14a) 2z k µ ik·X(0,0) ˜ µ (¯z ) : eik·X(0,0) : ∼ :, (2.3.14b) :e 2¯z from terms with a single contraction. This OPE is in agreement with the general identity (2.3.11). Another example is the world-sheet translation δσ a = v a , under which δX µ = −v a ∂a X µ . The Noether current is j µ (z) : eik·X(0,0) : ∼

Here Tab

ja = iv b Tab , 1 1 Tab = − : ∂a X µ ∂b Xµ − δab ∂c X µ ∂c Xµ : . α 2 is the world-sheet energy-momentum tensor.2 2.4

(2.3.15a) (2.3.15b)

Conformal invariance

The energy-momentum tensor (2.3.15b) is traceless, Taa = 0. In complex coordinates this is Tz¯z = 0 . The conservation

∂a Tab

(2.4.1)

= 0 then implies that in any theory with Taa = 0, ¯ zz = ∂T¯z¯z = 0 . ∂T (2.4.2)

Thus T (z) ≡ Tzz (z) , 2

˜ (¯z ) ≡ T¯z¯z (¯z ) T

(2.4.3)

In Tab we have used normal ordering to deﬁne the product of operators at a point. The only possible ambiguity introduced by the renormalization is a constant times δab , from the subtraction. Adding such a constant gives a diﬀerent energy-momentum tensor which is also conserved. We choose to focus on the tensor (2.3.15b), for reasons that will be explained more fully in the next chapter.

44

2 Conformal ﬁeld theory

are respectively holomorphic and antiholomorphic. For the free massless scalar, 1 ¯ µ: , ¯ µ ∂X ˜ (¯z ) = − 1 : ∂X T (z) = − : ∂X µ ∂Xµ : , T (2.4.4) α α which are indeed holomorphic and antiholomorphic as a consequence of the equation of motion. The tracelessness of Tab implies a much larger symmetry. The currents ˜ (¯z ) j(z) = iv(z)T (z) , ˜ (¯z ) = iv(z)∗ T (2.4.5) are conserved for any holomorphic v(z). For the free scalar theory, one ﬁnds the OPE 1 ¯ µ (0) . ˜ (¯z )X µ (0) ∼ 1 ∂X (2.4.6) T (z)X µ (0) ∼ ∂X µ (0) , T ¯z z The Ward identity then gives the transformation ¯ µ. (2.4.7) δX µ = −v(z)∂X µ − v(z)∗ ∂X This is an inﬁnitesimal coordinate transformation z = z + v(z). The ﬁnite transformation is X µ (z , ¯z ) = X µ (z, ¯z ) ,

z = f(z)

(2.4.8)

for any holomorphic f(z). This is known as a conformal transformation. The conformal symmetry we have found should not be confused with the diﬀ invariance of general relativity. We are in the ﬂat space theory, with no independent metric ﬁeld to vary, so the transformation z → z actually changes the distances between points. We would not ordinarily have such an invariance; it is a nontrivial statement about the dynamics. For the scalar action (2.1.10), the conformal transformation of ∂ and ∂¯ just balances that of d2 z. A mass term m2 X µ Xµ would not be invariant. Obviously there will in the end be a close relation with the diﬀ×Weyl symmetry of the Polyakov string, but we leave that for the next chapter. Consider the special case z = ζz

(2.4.9)

for complex ζ. The phase of ζ is a rotation of the system, while its magnitude is a rescaling of the size of the system. Such a scale invariance has occasionally been considered as an approximate symmetry in particle physics, and statistical systems at a critical point are described by scaleinvariant ﬁeld theories. To get some insight into the general conformal transformation, consider its eﬀect on inﬁnitesimal distances ds2 = dσ a dσ a = dzd¯z . Conformal transformations rescale this by a position-dependent factor, ds2 = dz d¯z =

∂z ∂¯z dzd¯z . ∂z ∂¯z

(2.4.10)

2.4 Conformal invariance

(a)

(b)

45

(c)

Fig. 2.2. (a) Two-dimensional region. (b) Eﬀect of the special conformal transformation (2.4.9). (c) Eﬀect of a more general conformal transformation.

Thus, as indicated in ﬁgure 2.2, a conformal transformation takes inﬁnitesimal squares into inﬁnitesimal squares, but rescales them by a positiondependent factor. An antiholomorphic function z = f(z)∗ has the same property, but changes the orientation. Most systems that are invariant under the rigid scaling (2.4.9) are actually invariant under the much larger conformal symmetry. A theory with this invariance is called a conformal ﬁeld theory (CFT). Conformal invariance in more than two dimensions is developed in exercise 2.6.

Conformal invariance and the OPE Conformal invariance puts strong constraints on the form of the OPE, and in particular on the OPEs of the energy-momentum tensor. Consider ˜ (¯z ) the OPE of T with the general operator A. Because T (z) and T are (anti)holomorphic except at insertions, the corresponding coeﬃcient functions must also have this property. The OPE of T with a general A is therefore a Laurent expansion, in integer but possibly negative powers of z. Further, all the singular terms are determined by the conformal transformation of A. To see this, let us write a general expansion of the singular terms, T (z)A(0, 0) ∼

∞ n=0

1 z n+1

A(n) (0, 0)

(2.4.11)

˜ ; the operator coeﬃcients A(n) remain to be determined. and similarly for T Under an inﬁnitesimal conformal transformation z = z + v(z), a single pole in v(z)T (z)A(0, 0) arises when the z −n−1 term of the T A OPE multiplies the term of order z n in v(z). Thus, the Ward identity in the

46

2 Conformal ﬁeld theory

form (2.3.11) implies that δA(z, ¯z ) = −

∞ 1 n ˜ (n) (z, ¯z ) . ∂ v(z)A(n) (z, ¯z ) + ∂¯n v(z)∗ A n=0

n!

(2.4.12)

Thus the operators A(n) are determined by the conformal transformation of A. It is convenient to take a basis of local operators that are eigenstates under rigid transformation (2.4.9), ˜

A (z , ¯z ) = ζ −h ¯ζ −h A(z, ¯z ) .

(2.4.13)

The (h, ˜h) are known as the weights of A. The sum h + h˜ is the dimension of A, determining its behavior under scaling, while h − ˜h is the spin, determining its behavior under rotations. The derivative ∂z increases h by one, and the derivative ∂¯z increases ˜h by one. The Ward identities for the transformation (2.4.13) and for the translation δA = −v a ∂a A determine part of the OPE, T (z)A(0, 0) = . . . +

1 h A(0, 0) + ∂A(0, 0) + . . . , 2 z z

(2.4.14)

˜. and similarly for T An important special case is a tensor operator or primary ﬁeld 3 O, on which a general conformal transformation acts as ˜

O (z , ¯z ) = (∂z z )−h (∂¯z ¯z )−h O(z, ¯z ) .

(2.4.15)

The OPE (2.4.11) reduces to T (z)O(0, 0) =

h 1 O(0, 0) + ∂O(0, 0) + . . . , 2 z z

(2.4.16)

the more singular terms in the general OPE (2.4.14) being absent. Taking again the example of the free X µ CFT, the weights of some typical operators are Xµ ¯ µ ∂X

(0, 0) , ∂X µ (1, 0) , (0, 1) , ∂2 X µ (2, 0) , 2 2 αk αk : eik·X : . , 4 4

3

(2.4.17)

In quantum ﬁeld theory, one usually distinguishes the fundamental ﬁelds (the variables of integration in the path integral) from more general operators which may be composite. Actually, this distinction is primarily useful in weakly coupled ﬁeld theories in three or more dimensions. It is of little use in CFT, in particular because of equivalences between diﬀerent ﬁeld theories, and so the term ﬁeld is used for any local operator.

47

2.4 Conformal invariance

All transform as tensors except ∂2 X µ . More generally, an exponential times a general product of derivatives, :

i∂

m i X µi

j

∂¯nj X νj eik·X : ,

(2.4.18)

has weight 2 αk

4

+

α k 2 + i mi , 4

j

nj

.

(2.4.19)

For any pair of operators, applying rigid translations, scale transformations, and rotations to both sides of the OPE determines the z-dependence of the coeﬃcient functions completely, Ai (z1 , ¯z1 )Aj (z2 , ¯z2 ) =

hk −hi −hj ˜hk −˜hi −˜hj ¯z12 z12 ckij Ak (z2 , ¯z2 ) ,

(2.4.20)

k

where the ckij are now constants. In all cases of interest the weights appearing on the right-hand side of the OPE (2.4.20) are bounded below, and so the degree of singularity in the operator product is bounded. More general conformal transformations put further constraints on the OPE: they determine the OPEs of all ﬁelds in terms of those of the primary ﬁelds. We will develop this in chapter 15. Notice that the conformal transformation properties of normal ordered products are not in general given by the naive transformations of the product. For example, the transformation law (2.4.7) for X µ would naively imply ¯ ik·X (naive) (2.4.21) δeik·X = −v(z)∂eik·X − v(z)∗ ∂e for the exponential, making it a tensor of weight (0, 0). The modiﬁcation is a quantum eﬀect, due to the renormalization needed to deﬁne the product of operators at a point. Speciﬁcally it enters here because the subtraction ln |z12 |2 in : : makes explicit reference to the coordinate frame. Conformal properties of the energy-momentum tensor The OPE of the energy-momentum tensor with itself was obtained in eq. (2.2.11), η µµ 2 − : ∂X µ (z)∂Xµ (0) : + : T (z)T (0) : 2z 4 α z 2 D 2 1 ∼ 4 + 2 T (0) + ∂T (0) . (2.4.22) 2z z z ˜ . The T (z)T ˜ (¯z ) OPE must be nonsingular. A similar result holds for T It cannot have poles in (z − z ) because it is antiholomorphic in z at nonzero separation; similarly it cannot have poles in (¯z − ¯z ) because it is T (z)T (0) =

48

2 Conformal ﬁeld theory

holomorphic in z at nonzero separation.4 The same holds for any OPE between a holomorphic and an antiholomorphic operator. Thus T is not a tensor. Rather, the OPE (2.4.22) implies the transformation law D (2.4.23) −1 δT (z) = − ∂z3 v(z) − 2∂z v(z)T (z) − v(z)∂z T (z) . 12 In a general CFT, T (z) transforms as c −1 δT (z) = − ∂z3 v(z) − 2∂z v(z)T (z) − v(z)∂z T (z) , (2.4.24) 12 with c a constant known as the central charge. The central charge of a free scalar is 1; for D free scalars it is D. The transformation (2.4.24) is the most general form that is linear in v, is consistent with the symmetry of the T T OPE, and has three lower z indices as required by rigid scale and rotation invariance. The scale, rotation, and translation symmetries determine the coeﬃcients of the second and third terms. Further, by considering the commutator of two such transformations one can show that ∂a c = 0, and it is a general result in quantum ﬁeld theory that an operator that is independent of position must be a c-number.5 The corresponding T T OPE is 2 1 c T (z)T (0) ∼ 4 + 2 T (0) + ∂T (0) . (2.4.25) 2z z z The ﬁnite form of the transformation law (2.4.24) is c (2.4.26) (∂z z )2 T (z ) = T (z) − {z , z} , 12 where {f, z} denotes the Schwarzian derivative {f, z} =

2∂z3 f∂z f − 3∂z2 f∂z2 f . 2∂z f∂z f

(2.4.27)

One can check this by verifying that it has the correct inﬁnitesimal form and that it composes correctly under successive transformations (so that one can integrate the inﬁnitesimal transformation). The corresponding ˜ , possibly with a diﬀerent central charge ˜c in the general forms hold for T CFT. The nontensor behavior of the energy-momentum tensor has a number of important physical consequences, as we will see. We should empha4

5

Unless otherwise stated OPEs hold only at nonzero separation, ignoring possible delta functions. For our applications the latter will not matter. Occasionally it is useful to include the delta functions, but in general these depend partly on deﬁnitions so one must be careful. The argument is this. Suppose that A(σ) is independent of position, and consider its equal time commutator with any other local operator B(σ ). These commute at spacelike separation, by locality. Since A(σ) is actually independent of position it also commutes at zero separation. Therefore A(σ) commutes with all local operators, and so must be a c-number.

2.5 Free CFTs

49

size that ‘nontensor’ refers to conformal transformations. The energymomentum tensor will have its usual tensor property under coordinate transformations. 2.5

Free CFTs

In this section we discuss three families of free-ﬁeld CFTs — the linear dilaton, bc, and βγ theories. The bc theory is the one of most immediate interest, as it will appear in the next chapter when we gauge-ﬁx the Polyakov string, but all have a variety of applications in string theory. Linear dilaton CFT This family of CFTs is based on the same action (2.1.10) but with energymomentum tensor 1 (2.5.1a) T (z) = − : ∂X µ ∂Xµ : +Vµ ∂2 X µ , α ¯ µ : +Vµ ∂¯2 X µ , ¯ µ ∂X ˜ (¯z ) = − 1 : ∂X T (2.5.1b) α where Vµ is some ﬁxed D-vector. Working out the T T OPE, one ﬁnds that it is of the standard form (2.4.25), but with central charge c = ˜c = D + 6α Vµ V µ .

(2.5.2)

The T X µ OPE and the Ward identity (2.4.12) imply the conformal transformation ¯ µ − α V µ [∂v + (∂v)∗ ] . (2.5.3) δX µ = −v∂X µ − v ∗ ∂X 2 This is a diﬀerent conformal symmetry of the same action. The ﬁeld X µ no longer transforms as a tensor, its variation now having an inhomogeneous piece. Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry — much more than we will have occasion to mention. The energy-momentum tensor plays a special role in string theory — in particular, it tells us how to couple to a curved metric — so diﬀerent values of V µ are to be regarded as diﬀerent CFTs. The vector V µ picks out a direction in spacetime. This CFT is therefore not Lorentz-invariant and not of immediate interest to us. We will see one physical interpretation of this linear dilaton CFT in section 3.7, and encounter it in some technical applications later. A diﬀerent variation of the free scalar CFT is to take some of the X µ to be periodic; we will take this up in chapter 8.

50

2 Conformal ﬁeld theory bc CFT

The second family of CFTs has anticommuting ﬁelds b and c with action 1 ¯ . S= d2 z b∂c (2.5.4) 2π This is conformally invariant for b and c transforming as tensors of weights (hb , 0) and (hc , 0) such that hb = λ ,

hc = 1 − λ

(2.5.5)

for any given constant λ. Thus we have another family of CFTs (which is secretly the same as the linear dilaton family, as we will learn in chapter 10). The operator equations of motion, obtained by the same method (2.1.15), (2.1.18) as before, are ¯ ¯ ∂c(z) = ∂b(z) =0, (2.5.6a) 2 ¯ (2.5.6b) ∂b(z)c(0) = 2πδ (z, ¯z ) . The bb and cc OPEs satisfy the equation of motion without source. The normal ordered bc product is 1 : b(z1 )c(z2 ) : = b(z1 )c(z2 ) − . (2.5.7) z12 This satisﬁes the naive equations of motion as a consequence of 1 1 ∂¯ = ∂ = 2πδ 2 (z, ¯z ) , (2.5.8) ¯z z which can be veriﬁed by integrating over a region containing the origin and integrating the derivative by parts. Normal ordering of a general product of ﬁelds is combinatorially the same as for the X µ CFT, a sum over contractions or subtractions. One must be careful because b and c are anticommuting so that interchange of ﬁelds ﬂips the sign: one should anticommute the ﬁelds being paired until they are next to each other before replacing them with the subtraction (2.5.7). The operator products are 1 1 , c(z1 )b(z2 ) ∼ , (2.5.9) b(z1 )c(z2 ) ∼ z12 z12 where in the second OPE there have been two sign ﬂips, one from anticommutation and one from z1 ↔ z2 . Other operator products are nonsingular: b(z1 )b(z2 ) = O(z12 ) ,

c(z1 )c(z2 ) = O(z12 ) .

(2.5.10)

These are not only holomorphic but have a zero due to antisymmetry. Noether’s theorem gives the energy-momentum tensor T (z) = : (∂b)c : −λ∂(: bc :) , ˜ (¯z ) = 0 . T

(2.5.11a) (2.5.11b)

51

2.5 Free CFTs

One can also verify (2.5.11) by working out the OPE of T with b and c; it has the standard tensor form (2.4.16) with the given weights. The T T OPE is of standard form (2.4.25) with c = −3(2λ − 1)2 + 1 ,

˜c = 0 .

(2.5.12)

This is a purely holomorphic CFT, and is an example where c = ˜c. There is of course a corresponding antiholomorphic theory

1 d2 z ˜b∂˜c , (2.5.13) 2π which is the same as the above with z ↔ ¯z . The bc theory has a ghost number symmetry δb = −ib, δc = ic. The corresponding Noether current is S=

j = − : bc : .

(2.5.14)

Again the components are separately holomorphic and antiholomorphic, the latter vanishing. When there are both holomorphic and antiholomorphic bc ﬁelds, the ghost numbers are separately conserved. This current is not a tensor, 1 1 1 − 2λ + 2 j(0) + ∂j(0) . z3 z z This implies the transformation law T (z)j(0) ∼

(2.5.15)

2λ − 1 2 ∂ v, 2

(2.5.16)

2λ − 1 ∂z2 z . 2 ∂z z

(2.5.17)

−1 δj = −v∂j − j∂v + whose ﬁnite form is (∂z z )jz (z ) = jz (z) +

The one case where b and c have equal weight is hb = hc = 12 , for which the central charge c = 1. Here we will often use the notation b → ψ, c → ψ. For this case the bc CFT can be split in two in a conformally invariant way, ψ = 2−1/2 (ψ1 + iψ2 ) , ψ = 2−1/2 (ψ1 − iψ2 ) , (2.5.18a) 1 ¯ 1 + ψ2 ∂ψ ¯ 2 , S = d2 z ψ1 ∂ψ (2.5.18b) 4π 1 1 T = − ψ1 ∂ψ1 − ψ2 ∂ψ2 . (2.5.18c) 2 2 Each ψ theory has central charge 12 . The bc theory for λ = 2, weights (hb , hc ) = (2, −1), will arise in the next chapter as the Faddeev–Popov ghosts from gauge-ﬁxing the Polyakov string. The ψ theory will appear extensively in the more general string theories of volume two.

52

2 Conformal ﬁeld theory βγ CFT

The third family of CFTs is much like the bc theory but with commuting ﬁelds; β is an (hβ , 0) tensor and γ an (hγ , 0) tensor, where hβ = λ , The action is

hγ = 1 − λ .

(2.5.19)

1 ¯ . d2 z β ∂γ 2π These ﬁelds are again holomorphic by the equations of motion, ¯ ¯ ∂γ(z) = ∂β(z) =0. S=

(2.5.20)

(2.5.21)

The equations of motion and operator products are derived in the standard way. Because the statistics are changed, some signs in operator products are diﬀerent, 1 1 , γ(z1 )β(z2 ) ∼ . (2.5.22) β(z1 )γ(z2 ) ∼ − z12 z12 The energy-momentum tensor is T = : (∂β)γ : −λ∂(: βγ :) , ˜ =0. T

(2.5.23a) (2.5.23b)

The central charge is simply reversed in sign, c = 3(2λ − 1)2 − 1 ,

˜c = 0 .

(2.5.24)

The βγ theory for λ = 32 , weights (hβ , hγ ) = ( 32 , − 12 ), will arise in chapter 10 as the Faddeev–Popov ghosts from gauge-ﬁxing the superstring. 2.6

The Virasoro algebra

Thus far in this chapter we have studied local properties of the twodimensional ﬁeld theory. We now are interested in the spectrum of the theory. The spatial coordinate will be periodic, as in the closed string, or bounded, as in the open string. For the periodic case let σ 1 ∼ σ 1 + 2π .

(2.6.1)

Let the Euclidean time coordinate run − ∞ < σ2 < ∞

(2.6.2)

so that the two dimensions form an inﬁnite cylinder. It is again useful to form a complex coordinate, and there are two natural choices. The ﬁrst is w = σ 1 + iσ 2 ,

(2.6.3)

2.6 The Virasoro algebra

w

53

z

2π

(a) 0

(b)

Fig. 2.3. Closed string coordinates. (a) Equal time contours in the w-plane. The dashed lines are identiﬁed. (b) The same contours in the z-plane.

so that w ∼ w + 2π. The second is z = exp(−iw) = exp(−iσ 1 + σ 2 ) .

(2.6.4)

These two coordinate systems are shown in ﬁgure 2.3. In terms of the w coordinate, time corresponds to translation of σ 2 = Im w. In terms of z, time runs radially, the origin being the inﬁnite past. These coordinates are related by a conformal transformation. The w coordinate is natural for the canonical interpretation of the theory, but the z coordinate is also quite useful and most expressions are written in this frame. For a holomorphic or antiholomorphic operator we can make a Laurent expansion, Tzz (z) =

∞

Lm , m+2 m=−∞ z

˜¯z¯z (¯z ) = T

∞

˜m L . z m+2 m=−∞ ¯

(2.6.5)

The Laurent coeﬃcients, known as the Virasoro generators, are given by the contour integrals

Lm =

C

dz m+2 Tzz (z) , z 2πiz

(2.6.6)

where C is any contour encircling the origin counterclockwise. The Laurent expansion is just the same as an ordinary Fourier transformation in the w frame at time σ 2 = 0: Tww (w) = −

∞

exp(imσ 1 − mσ 2 )Tm ,

m=−∞ ∞

¯ =− Tw¯ w¯ (w)

˜m , exp(−imσ 1 − mσ 2 )T

m=−∞

(2.6.7a) (2.6.7b)

54

2 Conformal ﬁeld theory

where c ˜m = L ˜ m − δm,0 ˜c . , T (2.6.8) 24 24 The additive shift of T0 is from the nontensor transformation (2.4.26), c Tww = (∂w z)2 Tzz + . (2.6.9) 24 Tm = Lm − δm,0

The Hamiltonian H of time translation in the w = σ 1 + iσ 2 frame is H=

2π dσ 1 0

˜ 0 − c + ˜c . T22 = L0 + L 2π 24

(2.6.10)

Notice that the +2s from the Laurent expansions (2.6.5) have canceled in the Fourier expansions (2.6.7) due to the conformal transformation of T . Similarly the Laurent expansion for a holomorphic ﬁeld of weight h would include +h in the exponent. Cutting open the path integral on circles of constant time Im w = ln |z|, the Virasoro generators become operators in the ordinary sense. (This idea of cutting open a path integral is developed in the appendix.) By holomorphicity the integrals (2.6.6) are independent of C and so in particular are invariant under time translation (radial rescaling). That is, they are conserved charges, the charges associated with the conformal transformations. It is an important fact that the OPE of currents determines the algebra of the corresponding charges. Consider general charges Qi , i = 1, 2, given as contour integrals of holomorphic currents, Qi {C} =

C

dz ji . 2πi

(2.6.11)

Consider the combination Q1 {C1 }Q2 {C2 } − Q1 {C3 }Q2 {C2 } ,

(2.6.12)

where the contours are shown in ﬁgure 2.4(a). The order in which the factors are written is irrelevant, as these are just variables of integration in a path integral (unless both charges are anticommuting, in which case there is an additional sign and all the commutators become anticommutators). As discussed in the appendix, when we slice open the path integral to make an operator interpretation, what determines the operator ordering is the time ordering, which here is t1 > t2 > t3 . The path integral with the combination (2.6.12) thus corresponds to a matrix element of ˆ 1Q ˆ 2Q ˆ 1, Q ˆ 2] . ˆ2 −Q ˆ 1 ≡ [Q Q

(2.6.13)

Now, for a given point z2 on the contour C2 , we can deform the diﬀerence of the C1 and C3 contours as shown in ﬁgure 2.4(b), so the commutator

55

2.6 The Virasoro algebra

z=0 C3

C2

C1

C2 z2 C1 C3

(a)

(b)

Fig. 2.4. (a) Contours centered on z = 0. (b) For given z2 on contour C2 , contour C1 − C3 is contracted.

is given by the residue of the OPE, [Q1 , Q2 ]{C2 } =

C2

dz2 Resz1 →z2 j1 (z1 )j2 (z2 ) . 2πi

(2.6.14)

This contour argument allows us to pass back and forth between OPEs and commutation relations. Let us emphasize this: for conserved currents, knowing the singular terms in the OPE is equivalent to knowing the commutator algebra of the corresponding charges. The calculation of ﬁgure 2.4 also applies with the conserved charge Q2 {C2 } replaced by any operator, 1 (2.6.15) δA(z2 , ¯z2 ) . i This is just the familiar statement that a charge Q generates the corresponding transformation δ. Similarly for the contour integral of an antiholomorphic current [Q, A(z2 , ¯z2 )] = Resz1 →z2 j(z1 )A(z2 , ¯z2 ) =

d¯z ˜ , 2πi C the Ward identity and contour argument imply ˜ Q{C} =−

(2.6.16)

1 (2.6.17) δA(z2 , ¯z2 ) . i Apply this to the Virasoro generators (2.6.6), where jm (z) = z m+1 T (z): ˜ A(z2 , ¯z2 )] = Res¯z →¯z ˜ (¯z1 )A(z2 , ¯z2 ) = [Q, 1 2

Resz1 →z2 z1m+1 T (z1 ) z2n+1 T (z2 )

=

Resz1 →z2 z1m+1 z2n+1

c 2 1 + 2 T (z2 ) + ∂T (z2 ) 4 z12 2z12 z12

56

2 Conformal ﬁeld theory c 3 m+1 n+1 (∂ z2 )z2 + 2(∂z2m+1 )z2n+1 T (z2 ) + z2m+n+2 ∂T (z2 ) 12 c 3 = (m − m)z2m+n−1 + (m − n)z2m+n+1 T (z2 ) + total derivative . 12 (2.6.18)

=

The z2 contour integral of the right-hand side then gives the Virasoro algebra, c [Lm , Ln ] = (m − n)Lm+n + (m3 − m)δm,−n . (2.6.19) 12 ˜ m satisfy the same algebra with central charge ˜c. The L Any CFT thus has an inﬁnite set of conserved charges, the Virasoro generators, which act in the Hilbert space and which satisfy the algebra (2.6.19). Let us for now notice just a few simple properties. Generally ˜ 0 . The generator L0 satisﬁes we work with eigenstates of L0 and L [L0 , Ln ] = −nLn .

(2.6.20)

If |ψ is an eigenstate of L0 with eigenvalue h, then L0 Ln |ψ = Ln (L0 − n)|ψ = (h − n)Ln |ψ ,

(2.6.21)

so that Ln |ψ is an eigenstate with eigenvalue h − n. The generators with n < 0 raise the L0 eigenvalue and those with n > 0 lower it. The three generators L0 and L±1 form a closed algebra without central charge, [L0 , L1 ] = −L1 ,

[L0 , L−1 ] = L−1 ,

[L1 , L−1 ] = 2L0 .

(2.6.22)

This is the algebra S L(2, R), which diﬀers from S U(2) by signs. For the Laurent coeﬃcients of a holomorphic tensor ﬁeld O of weight (h, 0), O(z) =

∞

Om , m+h z m=−∞

(2.6.23)

one ﬁnds from the OPE (2.4.16) the commutator [Lm , On ] = [(h − 1)m − n]Om+n .

(2.6.24)

Again modes with n > 0 reduce L0 , while modes with n < 0 increase it. In the open string, let 0 ≤ Re w ≤ π

⇔

Im z ≥ 0 ,

(2.6.25)

where z = − exp(−iw). These coordinate regions are shown in ﬁgure 2.5. At a boundary, the energy-momentum tensor satisﬁes Tab na tb = 0 ,

(2.6.26)

where na and tb are again normal and tangent vectors. To see this, consider a coordinate system in which the boundary is straight. The presence of

2.6 The Virasoro algebra

57

w z

(a) 0

π

(b)

Fig. 2.5. Open string coordinates. (a) Equal time contours in the w-plane. (b) The same contours in the z-plane. The dashed line shows the extension of one contour, as used in the doubling trick.

the boundary breaks translation invariance in the normal direction but not the tangential, so that the current Tab tb is still conserved. Then the boundary condition (2.6.26) is just the statement that the ﬂow of this current out of the boundary is zero. In the present case, this becomes Tww = Tw¯ w¯ , Re w = 0, π

⇔

Tzz = T¯z¯z , Im z = 0 .

(2.6.27)

It is convenient to use the doubling trick. Deﬁne Tzz in the lower halfz-plane as the value of T¯z¯z at its image in the upper half-z-plane, z = ¯z : Tzz (z) ≡ T¯z¯z (¯z ) ,

Imz < 0 .

(2.6.28)

The equation of motion and boundary condition are then summarized by the statement that Tzz is holomorphic in the whole complex plane. There is only one set of Virasoro generators, because the boundary condition ˜, couples T and T

1 dz z m+1 Tzz − d¯z ¯z m+1 T¯z¯z Lm = 2πi C 1 = dz z m+1 Tzz (z) . 2πi

(2.6.29)

In the ﬁrst line, the contour C is a semi-circle centered on the origin; in the second line, we have used the doubling trick to write Lm in terms of a closed contour. Again, these satisfy the Virasoro algebra [Lm , Ln ] = (m − n)Lm+n +

c 3 (m − m)δm,−n . 12

(2.6.30)

58

2 Conformal ﬁeld theory 2.7

Mode expansions Free scalars

In free ﬁeld theory, the ﬁelds break up into harmonic oscillators, and the spectrum and energy-momentum tensor can be given in terms of the ¯ are modes. We start with the closed string. In the X µ theory, ∂X and ∂X (anti)holomorphic and so have Laurent expansions like that for T , ∂X µ (z) = −i

1/2 ∞ α

2

αµm

m=−∞

z m+1

¯ µ (¯z ) = −i α ∂X 2

,

1/2 ∞

˜αµm . z m+1 m=−∞ ¯ (2.7.1)

Equivalently,

2 α

1/2

dz m µ z ∂X (z) , 2π 1/2 2 d¯z m ¯ µ ¯z ∂X (¯z ) . α˜µm = − α 2π αµm =

(2.7.2a) (2.7.2b)

Single-valuedness of X µ implies that αµ0 = α˜µ0 . Moreover, the Noether current for spacetime translations is i∂a X µ /α , so the spacetime momentum is 1 p = 2πi

µ

(dz j − d¯z ˜ ) = µ

C

µ

2 α

1/2

αµ0

=

2 α

1/2

˜αµ0 .

(2.7.3)

Integrating the expansions (2.7.1) gives

α α X (z, ¯z ) = x − i pµ ln |z|2 + i 2 2 µ

µ

1/2 ∞ m=−∞ m=0

1 αµm α˜µm + m z m ¯z m

.

(2.7.4)

Either from standard canonical commutation, or from the contour argument and the XX OPE, one derives [αµm , ανn ] = [˜αµm , ˜ανn ] = mδm,−n η µν ,

(2.7.5a)

[xµ , pν ] = iη µν ,

(2.7.5b)

with other commutators vanishing. The spectrum is given by starting with a state |0; k that has momentum k µ and is annihilated by all of the lowering modes, αµn for n > 0, and acting in all possible ways with the raising (n < 0) modes. We now wish to expand the Virasoro generators in terms of the mode operators. Insert the Laurent expansion for X µ into the energy-momentum

2.7 Mode expansions

59

tensor (2.4.4) and collect terms with a given power of z, giving Lm ∼

∞ 1 αµm−n αµn . 2 n=−∞

(2.7.6)

The ∼ indicates that we have ignored the ordering of operators. For m = 0, the expansion (2.7.6) is well deﬁned and correct as it stands — the mode operators in each term commute and so the ordering does not matter. For m = 0, put the lowering operators on the right and introduce a normal ordering constant, L0 =

∞ α p2 + (αµ−n αµn ) + aX . 4 n=1

(2.7.7)

We encountered this same issue in section 1.3, where we treated it in a heuristic way. Now the left-hand side has a ﬁnite and unambiguous deﬁnition in terms of the Laurent coeﬃcients of the : :-ordered energymomentum tensor, so the normal ordering constant is ﬁnite and calculable. There are several ways to determine it. The simplest uses the Virasoro algebra, 2L0 |0; 0 = (L1 L−1 − L−1 L1 )|0; 0 = 0 ,

(2.7.8)

aX = 0 .

(2.7.9)

and so Here we have used the known form of L1 and L−1 : every term of each contains either a lowering operator or pµ and so annihilates |0; 0 . We determined the central charge in the Virasoro algebra from the OPE. It can also be determined directly from the expression for the Virasoro generators in terms of the mode operators, though some care is needed. This is left as exercise 2.11. Let us introduce a new notation. The symbol ◦◦ ◦◦ will denote creation– annihilation normal ordering, placing all lowering operators to the right of all raising operators, with a minus sign whenever anticommuting operators are switched. For the purposes of this deﬁnition, we will include pµ with the lowering operators and xµ with the raising operators. In this notation we can write Lm =

∞ 1 ◦ µ α ◦◦ ◦α 2 n=−∞ m−n µn

(2.7.10)

since aX = 0. We have now introduced two forms of normal ordering, conformal normal ordering : : (which is what we will mean if we just refer to ‘normal ordering’) and creation–annihilation normal ordering ◦◦ ◦◦ . The former is useful because it produces operators whose OPEs and conformal

60

2 Conformal ﬁeld theory

transformation properties are simple. The latter, which is probably more familiar to the reader, is useful for working out the matrix elements of the operators. Let us work out the relation between them. We start by comparing the time-ordered and creation–annihilation-ordered products. For the product X µ (z, ¯z )X ν (z , ¯z ) with |z| > |z |, insert the mode expansions and move the lowering operators in X µ (z, ¯z ) to the right. Keeping track of the commutators gives X µ (z, ¯z )X ν (z , ¯z ) = ◦◦ X µ (z, ¯z )X ν (z , ¯z )◦◦

∞ 1 z m ¯z m α + η µν − ln |z|2 + + m ¯z 2 m zm m=1

= ◦◦ X µ (z, ¯z )X ν (z , ¯z )◦◦ −

α µν η ln |z − z |2 . 2

(2.7.11)

Since |z| > |z |, the left-hand side here is time-ordered and the sum converges. The deﬁnition (2.1.21) gave the relation between the timeordered product and the conformal-normal-ordered product (eq. (2.1.21) was in path integral language, so the product on its right becomes timeordered in operator formalism). It is the same as the relation (2.7.11), so that ◦ ◦

X µ (z, ¯z )X ν (z , ¯z )◦◦ = : X µ (z, ¯z )X ν (z , ¯z ) : .

(2.7.12)

Such a relation does not hold in general CFTs, and in fact the somewhat arbitrary grouping of pµ with the lowering operators was done in part to give this simple result. It also does not hold for operators conformalnormal-ordered in terms of w rather than z, for example. From eq. (2.7.12) one can write the mode expansion (2.7.10) at once, giving a second derivation of aX = 0. Creation–annihilation normal-ordered products of more than two X µ s have the same combinatoric properties as for : : ordering. That is, they are obtained from the time-ordered product by summing over all subtractions — this is Wick’s theorem from ﬁeld theory. Also, to convert operators normal-ordered in one form into a diﬀerent normal ordering, one sums over all subtractions using the diﬀerence of the two-point functions. That is, if we have two kinds of ordering, [X µ (z, ¯z )X ν (z , ¯z )]1 = [X µ (z, ¯z )X ν (z , ¯z )]2 + η µν ∆(z, ¯z , z , ¯z ) , then for a general operator F,

1 [F]1 = exp 2

(2.7.13)

δ δ d z d z ∆(z, ¯z , z , ¯z ) µ [F]2 . δX (z, ¯z ) δXµ (z , ¯z ) (2.7.14) For the linear dilaton CFT, the Laurent expansion and commutators 2

2

61

2.7 Mode expansions are unchanged, while the Virasoro generators contain the extra term

Lm =

∞ α 1 ◦ µ ◦ ◦ αm−n αµn ◦ + i 2 n=−∞ 2

1/2

(m + 1)V µ αµm .

(2.7.15)

bc CFT The ﬁelds b and c have the Laurent expansions ∞

bm

m=−∞

z m+λ

b(z) =

,

∞

c(z) =

m=−∞

cm m+1−λ z

.

(2.7.16)

To be precise, these are only Laurent expansions if λ is an integer, which we will assume for now. The half-integer case is also of interest, but will be dealt with in detail in chapter 10. The OPE gives the anticommutators {bm , cn } = δm,−n .

(2.7.17)

Consider ﬁrst the states that are annihilated by all of the n > 0 operators. The b0 , c0 oscillator algebra generates two such ground states | ↓ and | ↑ , with the properties b0 | ↓ = 0 , b0 | ↑ = | ↓ , c0 | ↓ = | ↑ , c0 | ↑ = 0 , bn | ↓ = bn | ↑ = cn | ↓ = cn | ↑ = 0 ,

n>0.

(2.7.18a) (2.7.18b) (2.7.18c)

The general state is obtained by acting on these states with the n < 0 modes at most once each (because these anticommute). For reasons to appear later it is conventional to group b0 with the lowering operators and c0 with the raising operators, so we will single out | ↓ as the ghost vacuum |0 . In string theory we will have a holomorphic bc theory and an antiholomorphic ˜b˜c theory, each with λ = 2. The closed string spectrum thus includes a product of two copies of the above. The Virasoro generators are ∞

Lm =

(mλ − n)◦◦ bn cm−n ◦◦ + δm,0 ag .

(2.7.19)

n=−∞

The ordering constant can be determined as in eq. (2.7.8), which gives 2L0 | ↓ = (L1 L−1 − L−1 L1 )| ↓

= (λb0 c1 )[(1 − λ)b−1 c0 ]| ↓ = λ(1 − λ)| ↓ .

(2.7.20)

Thus, ag = 12 λ(1 − λ) and Lm =

∞

(mλ − n)◦◦ bn cm−n ◦◦ +

n=−∞

λ(1 − λ) δm,0 . 2

(2.7.21)

62

2 Conformal ﬁeld theory

The constant can also be obtained by working out the relation between : : and ◦◦ ◦◦ for the CFT (exercises 2.13, 2.14). For the ghost number current (2.5.14), j = − : bc :, the charge is Ng = − =

1 2πi

∞

2π

dw jw

0

(c−n bn − b−n cn ) + c0 b0 −

n=1

1 . 2

(2.7.22)

It satisﬁes [N g , bm ] = −bm ,

[N g , cm ] = cm ,

(2.7.23)

and so counts the number of c minus the number of b excitations. The ground states have ghost number ± 12 : 1 1 N g | ↓ = − | ↓ , N g | ↑ = | ↑ . (2.7.24) 2 2 This depends on the value of the ordering constant, determined in exercise 2.13, but one might guess it on the grounds that the average ghost number of the ground states should be zero: the ghost number changes sign under b ↔ c. The βγ theory is similar; we leave the details until we need them in chapter 10. Open strings In the open string, the Neumann boundary condition becomes ∂z X µ = ∂¯z X µ on the real axis. There is only one set of modes, the boundary condition requiring αµm = ˜αµm in the expansions (2.7.1). The spacetime momentum integral (2.7.3) runs only over a semi-circle, so the normalization is now αµ0 = (2α )1/2 pµ .

(2.7.25)

The expansion for X µ is then µ

X (z, ¯z ) = x − iα p ln |z| + i µ

µ

2

1/2 ∞ α αµm

2

m=−∞ m=0

m

(z −m + ¯z −m ) .

(2.7.26)

Also, 2

L0 = α p +

∞ µ

α−n αµn .

(2.7.27)

[xµ , pν ] = iη µν .

(2.7.28)

n=1

The commutators are as before, [αµm , ανn ] = mδm,−n η µν ,

63

2.8 Vertex operators

For the bc theory, the boundary conditions that will be relevant to the string are c(z) = ˜c(¯z ) ,

b(z) = ˜b(¯z ) ,

Im z = 0 ,

(2.7.29)

written in terms of the z-coordinate where the boundary is the real axis. We can then use the doubling trick to write the holomorphic and antiholomorphic ﬁelds in the upper half-plane in terms of holomorphic ﬁelds in the whole plane, c(z) ≡ ˜c(¯z ) ,

b(z) ≡ ˜b(¯z ) ,

Im(z) ≤ 0 , z = ¯z .

(2.7.30)

The open string thus has a single set of Laurent modes for each of b and c. 2.8

Vertex operators

In quantum ﬁeld theory, there is on the one hand the space of states of the theory and on the other hand the set of local operators. In conformal ﬁeld theory there is a simple and useful isomorphism between these, with the CFT quantized on a circle. Consider the semi-inﬁnite cylinder in the w-coordinate, 0 ≤ Re w ≤ 2π ,

w ∼ w + 2π ,

Im w ≤ 0 ,

(2.8.1)

which maps into the unit disk in the coordinate z = exp(−iw). This is shown in ﬁgure 2.6. To deﬁne the path integral in the w-coordinate one must in particular specify the boundary condition as Im w → −∞. That is, one must specify the initial state. In the z-coordinate, Im w = −∞ maps to the origin, so this is equivalent to specifying the behavior of the ﬁelds at that point. In eﬀect, this deﬁnes a local operator at the origin, known as the vertex operator associated with the state. Going in the other direction, the path integral on the disk with an operator A at the origin maps to the path integral on the cylinder with a speciﬁed initial state |A . For free ﬁeld theories one can easily work out the detailed form of this isomorphism. Suppose one has a conserved charge Q acting on the state |A as in ﬁgure 2.6(a). One can ﬁnd the corresponding local operator by using the OPE to evaluate the contour integral in ﬁgure 2.6(b). Let us use this to identify the state |1 corresponding to the unit operator. With no ¯ µ are (anti)holomorphic inside the Q operator at the origin, ∂X µ and ∂X contour in ﬁgure 2.6(b). The contour integrals (2.7.2) deﬁning αµm and ˜αµm for m ≥ 0 then have no poles and so vanish. Thus |1 is annihilated by these modes. This identiﬁes it as the ground state, |1 = |0; 0 ,

(2.8.2)

64

2 Conformal ﬁeld theory

Q A

(a)

Q

(b)

A Fig. 2.6. (a) A semi-inﬁnite cylinder in w, with initial state |A and charge Q. (b) The conformally equivalent unit disk with local operator A and contour integral for Q.

with the normalization chosen for convenience. Now consider, for example, the state αµ−m |1 with m positive. Passing to ﬁgure 2.6(b) with Q = αµ−m surrounding the unit operator, the ﬁelds are holomorphic inside the contour, and so we can evaluate αµ−m =

2 α

1/2

dz −m µ z ∂X (z) → 2π

for m ≥ 1. Thus αµ−m |1 ∼ =

2 α

2 α

1/2

i ∂m X µ (0) (m − 1)!

1/2

i ∂m X µ (0) , (m − 1)!

(2.8.3)

m≥1.

(2.8.4)

m≥1.

(2.8.5)

Similarly ˜αµ−m |1

∼ =

2 α

1/2

i ∂¯m X µ (0) , (m − 1)!

This correspondence continues to hold when αµ−m or ˜αµ−m acts on a general state |A . If : A(0, 0) : is any normal-ordered operator, there may be singularities in the operator product of ∂X µ (z) with : A(0, 0) :, but it is not hard to check that αµ−m : A(0, 0) : = : αµ−m A(0, 0) :

(2.8.6)

65

2.8 Vertex operators

for m > 0 because the contour integral of the contractions will never have a single pole. One can then carry out the same manipulation (2.8.3) inside the normal ordering, and so the state with several α oscillators excited comes out as the : : normal-ordered product of the corresponding operators, αµ−m → i ˜ αµ−m → i

2 α 2 α

1/2

1 ∂m X µ (0) , (m − 1)!

m≥1,

(2.8.7a)

1 ∂¯m X µ (0) , (m − 1)!

m≥1.

(2.8.7b)

1/2

Similarly, xµ0 → X µ (0, 0) .

(2.8.8)

Any state can be obtained from |1 by acting with the operators on the left-hand sides of eqs. (2.8.7) and (2.8.8). The operator corresponding to this state is then given by the : : normal-ordered product of the corresponding local operators on the right-hand sides. For example, |0; k ∼ = : eik·X(0,0) : .

(2.8.9)

This is easy to understand: under the translation X µ → X µ + aµ , both the state and the corresponding operator have the same transformation, being multiplied by exp(ik · a). The same method applies to the bc theory. For clarity we specialize to the case λ = 2 which is of interest for bosonic string theory, leaving the generalization to the reader. From the Laurent expansions (2.7.16) and the contour argument it follows that bm |1 = 0 ,

m ≥ −1 ,

cm |1 = 0 ,

m≥2.

(2.8.10)

Note that due to the shifts in the exponents of the Laurent expansion coming from the conformal weights of b and c, the unit operator no longer maps to a ground state. Rather, relations (2.8.10) determine that |1 = b−1 | ↓ .

(2.8.11)

The translation of the raising operators is straightforward, 1 (2.8.12a) ∂m−2 b(0) , m ≥ 2 , (m − 2)! 1 (2.8.12b) c−m → ∂m+1 c(0) , m ≥ −1 . (m + 1)! Notice that the ghost number − 32 state b−1 | ↓ maps to the ghost number 0 unit operator, and the ghost number − 12 state | ↓ to the ghost number 1 operator c. The diﬀerence arises from the nontensor b−m →

66

2 Conformal ﬁeld theory

property (2.5.17) of the ghost number current, (∂z w)jw (w) = jz (z) + q0

q0 ∂z2 w = jz (z) − , ∂z w z

(2.8.13)

where q0 = λ − 12 = 32 . The ghost number of states is conventionally deﬁned by the cylindrical frame expression N g , eq. (2.7.22), while the contour argument of ﬁgure 2.6 relates the ghost number of the vertex operator to the radial frame charge

1 dz jz = N g + q0 . (2.8.14) 2πi This applies to other charges as well; for a tensor current the charges of a state and the corresponding operator are equal. Most of the above can be extended to the βγ theory, but there are certain complications in the superstring so we defer this to chapter 10. All the ideas of this section extend to the open string. The semi-inﬁnite strip Qg ≡

0 ≤ Re w ≤ π ,

Im w ≤ 0 ,

(2.8.15)

maps to a half-disk, the intersection of the upper half-plane and the unit circle, under z = − exp(−iw). The initial state again maps to the origin, which is on the boundary, so there is an isomorphism local operators on the boundary ↔ states on the interval .

(2.8.16)

The details, which are parallel to the above, are left for the reader. Again the doubling trick, extending the ﬁelds into the lower half-plane by holomorphicity, is useful in making the contour arguments. Path integral derivation The state–operator isomorphism is an important but unfamiliar idea and so it is useful to give also a more explicit path integral derivation. Consider a unit disk in the z-plane, with local operator A at the origin and with the path integral ﬁelds φ ﬁxed to some speciﬁc boundary values φb on the unit circle. The path integral over the ﬁelds φi on the interior of the disk with φb held ﬁxed produces some functional ΨA [φb ],

ΨA [φb ] =

[dφi ]φb exp(−S [φi ])A(0) .

(2.8.17)

A functional of the ﬁelds is the Schr¨ odinger representation of a state, so this is a mapping from operators to states. The Schr¨ odinger representation, which assigns a complex amplitude to each ﬁeld conﬁguration, has many uses. It is often omitted from ﬁeld theory texts, the Fock space representation being emphasized.

67

2.8 Vertex operators

To go the other way, start with some state Ψ[φb ]. Consider a path integral over the annular region 1 ≥ |z| ≥ r, with the ﬁelds φb on the outer circle ﬁxed and the ﬁelds φb integrated over the inner circle as follows:

˜

[dφb ][dφi ]φb ,φb exp(−S [φi ]) r−L0 −L0 Ψ[φb ] .

(2.8.18) ˜

That is, the integral is weighted by the state (functional) r−L0 −L0 Ψ[φb ]. Now, the path integral over the annulus just corresponds to propagating from |z| = r to |z| = 1, which is equivalent to acting with the operator ˜ r+L0 +L0 . This undoes the operator acting on Ψ, so the path integral (2.8.18) is again Ψ[φb ]. Now take the limit as r → 0. The annulus becomes a disk, and the limit of the path integral over the inner circle can be thought of as deﬁning some local operator at the origin. By construction the path integral on the disk with this operator reproduces Ψ[φb ] on the boundary. Let us work this out explicitly for a single free scalar ﬁeld X. On the unit circle expand Xb (θ) =

∞

Xn∗ = X−n .

Xn einθ ,

n=−∞

(2.8.19)

The boundary state Ψ[Xb ] can thus be regarded as a function of all of the Xn . Let us ﬁrst identify the state corresponding to the unit operator, given by the path integral without insertion,

Ψ1 [Xb ] =

[dXi ]Xb

1 exp − 2πα

¯ d z ∂X ∂X

2

(2.8.20)

.

Evaluate this by the usual Gaussian method. Separate Xi into a classical part and a ﬂuctuation, Xi = Xcl + Xi , Xcl (z, ¯z ) = X0 +

∞

(2.8.21a)

(z n Xn + ¯z n X−n ) .

(2.8.21b)

n=1

With this deﬁnition, Xcl satisﬁes the equation of motion and Xi vanishes on the boundary. The path integral then separates,

Ψ1 [Xb ] = exp(−Scl ) with

[dXi ]Xb =0

1 exp − 2πα

2

¯

d z ∂X ∂X

,

(2.8.22)

∞ 1 Scl = mnXm X−n d2 z z m−1 ¯z n−1 2πα m,n=1 |z|<1

=

∞ 1 mXm X−m . α m=1

(2.8.23)

68

2 Conformal ﬁeld theory

The Xi integral is a constant, independent of the boundary condition, so

∞ 1 Ψ1 [Xb ] ∝ exp − mXm X−m α m=1

.

(2.8.24)

This is a Gaussian, and is in fact the ground state. To see this, write the raising and lowering operators in the Schr¨ odinger basis,

α in αn = − 1/2 X−n − i 2 (2α )

1/2

∂ , ∂Xn

(2.8.25a)

α 1/2 ∂ in ˜αn = − 1/2 Xn − i , (2.8.25b) 2 ∂X−n (2α ) as follows from the Laurent expansion at |z| = 1 and the mode algebra. Acting on (2.8.24), we ﬁnd αn Ψ1 [Xb ] = ˜αn Ψ1 [Xb ] = 0 ,

n≥0,

(2.8.26)

so this is indeed the ground state. Thus, |1 ∝ |0; 0 .

(2.8.27)

Rather than try to keep track of the overall normalization of the path integral at this point, we deﬁne |1 = |0; 0 . Another easy calculation is the state corresponding to ∂k X; this just adds a factor of ∂k Xcl (0) = k!Xk to the result, so |∂k X = k!Xk Ψ1 = −i

1/2 α

2

(k − 1)!α−k |0; 0 ,

(2.8.28)

and similarly for ∂¯k X. This extends to all products of exponentials and derivatives of X. The conformal normal ordering just cancels the eﬀect of the Xi path integral. 2.9

More on states and operators The OPE

In this section we make additional applications of the state–operator correspondence. The ﬁrst is a simple and general derivation of the OPE, as shown in ﬁgure 2.7. Consider the product Ai (z, ¯z )Aj (0, 0) ,

(2.9.1)

where |z| < 1. We can divide the path integral into an integral over ﬁelds φi on the interior of the unit disk, an integral over ﬁelds φb on the unit circle, and an integral over ﬁelds φe on the exterior of the unit disk. This sort of cutting open of path integrals is discussed in detail in the

2.9 More on states and operators

69

A i(z,z)

ψij,z,z

A j(0,0)

(b)

(a)

A ij,z,z

(c) Fig. 2.7. (a) World-sheet with two local operators. (b) Integration over ﬁelds on the interior of the disk produces boundary state |ψij,z,¯z . (c) Sewing in a disk with the corresponding local operator. Expanding in operators of deﬁnite weight gives the OPE.

appendix. As in the discussion at the end of the previous section, the integral over φi leaves some functional of φb ; call it Ψij,z,¯z [φb ]. By the state–operator correspondence this is equivalent to gluing back the disk with the appropriate local operator Aij,z,¯z (0, 0) and so we are done! To ˜ 0 eigenstates, put this in standard form expand in a complete set of L0 , L Aij,z,¯z =

˜

˜

˜

z hk −hi −hj ¯z hk −hi −hj ckij Ak ,

(2.9.2)

k

with the z- and ¯z -dependence determined by the conformal weights as in eq. (2.4.20). The convergence of the OPE is just the usual convergence of a complete set in quantum mechanics. The construction is possible as long as there are no other operators with |z | ≤ |z|, so that we can cut on a circle of radius |z| + .

70

2 Conformal ﬁeld theory m

i i

Σl

m

=

l j

k

Σl

l

k

j

Fig. 2.8. Schematic picture of OPE associativity.

For three operators Ai (0, 0)Aj (1, 1)Ak (z, ¯z ) ,

(2.9.3)

the regions of convergence of the z → 0 and z → 1 OPEs, |z| < 1 and |1 − z| < 1, overlap. The coeﬃcient of Am in the triple product can then be written as a sum involving clik cmlj or as a sum involving cljk cmli . Associativity requires these sums to be equal. This is represented schematically in ﬁgure 2.8. The Virasoro algebra and highest weight states Consider now the argument of ﬁgure 2.6 for the Virasoro generators:

dz m+1 Lm |A ∼ z T (z)A(0, 0) = 2πi ∼ = Lm · A(0, 0) .

(2.9.4)

Here we have introduced a new idea and notation. Given the state– operator isomorphism, every operator acting on the Hilbert space has an image acting on the space of local operators. In other words, form the corresponding contour around the operator and evaluate the resulting local operator using the OPE. Corresponding to the Lm that map states to states are the Lm · that map local operators to local operators. In general there will be local operators at various positions zi , and there will be a diﬀerent copy of the Virasoro algebra for each one, obtained by a Laurent expansion centered on the position of the operator. In some geometries there may also be a standard set of generators, such as the ones centered on z = 0 for the cylinder. The ‘·’ serves as a reminder that we are talking about the Laurent coeﬃcients around that particular operator. In this notation the T A OPE is T (z)A(0, 0) =

∞ m=−∞

z −m−2 Lm · A(0, 0) .

(2.9.5)

2.9 More on states and operators

71

To relate to the earlier notation (2.4.11), A(n) = Ln−1 · A for n ≥ 0, and the conformal transformation of A is δA(z, ¯z ) = −

∞ 1 n ˜ n−1 · A(z, ¯z ) . ∂ v(z)Ln−1 + (∂n v(z))∗ L n=0

n!

(2.9.6)

Using the general form (2.4.14) for the T A OPE, we have the very useful result ¯ , ˜ −1 · A = ∂A L−1 · A = ∂A , L ˜ 0 · A = ˜hA . L0 · A = hA , L

(2.9.7a) (2.9.7b)

The OPE (2.9.5) implies that a primary ﬁeld O of weight (h, ˜h) corresponds to a state satisfying ˜ 0 |O = ˜h|O , L0 |O = h|O , L ˜ m |O = 0 , m > 0 . Lm |O = L

(2.9.8a) (2.9.8b)

Such a state is conventionally known as a highest weight state. For the ˜ 0 are bounded below. By acting repeatedly with CFTs of interest L0 and L lowering operators on any state we eventually reach a state annihilated by further lowering operators, a highest weight state. An interesting special case is the unit operator. The operator product T 1 is nonsingular, so it follows in any CFT that ˜ m |1 = 0 , Lm |1 = L

m ≥ −1 .

(2.9.9)

As noted in section 2.6, the operators L0 and L±1 form a closed algebra, ˜ ±1 . The full algebra is ˜ 0 and L as do L S L(2, C) .

(2.9.10)

Thus |1 is also called the S L(2, C)-invariant state. It is the only such state because the relations (2.9.7) imply that the operator A corresponding to any S L(2, C)-invariant state is independent of position. It must then be a c-number, as explained after eq. (2.4.24). Unitary CFTs Highest weight states play a special role in string theory, and also in the theory of representations of the Virasoro algebra. We will return to this at various points, but for now we derive a few important general results that hold in unitary CFTs. A unitary CFT is one that has a positive inner product | such that L†m = L−m ,

˜ †m = L ˜ −m . L

(2.9.11)

Recall that the inner product deﬁnes the adjoint via α|Aβ = A† α|β .

72

2 Conformal ﬁeld theory

For example, the X µ CFT is unitary for spacelike µ if we take the inner product

0; k|0; k = 2πδ(k − k )

(2.9.12)

for the ground state and deﬁne α†m = α−m ,

˜α†m = ˜α−m .

(2.9.13)

This implicitly deﬁnes the inner products of all higher states. This CFT is not unitary for µ = 0 because of the opposite sign in the commutator there. The ﬁrst constraint is that any state in a unitary CFT must have h, ˜h ≥ 0. If this is true for highest weight states it is true for all states. For a highest weight state the Virasoro algebra gives 2hO O|O = 2 O|L0 |O = O|[L1 , L−1 ]|O = L−1 |O 2 ≥ 0 ,

(2.9.14)

so hO ≥ 0. It also follows that if hO = ˜hO = 0 then ˜ −1 · O = 0 , L−1 · O = L

(2.9.15)

and so O is independent of position. As noted before, O must then be a c-number. That is, the unit operator is the only (0,0) operator in a unitary CFT. Curiously, X µ itself is the one notable exception: the corresponding state xµ |0; 0 is nonnormalizable because of the inﬁnite range of X µ , and equation (2.9.14) no longer holds. The general theorems are of the most use for the CFTs corresponding to compactiﬁed dimensions, where this kind of exception cannot occur. In a similar way, one ﬁnds that an operator in a unitary CFT is holomorphic if and only if ˜h = 0, and antiholomorphic if and only if h = 0; this is an important result so we repeat it, ∂A = 0 ⇔ h = 0 ,

¯ = 0 ⇔ ˜h = 0 . ∂A

(2.9.16)

Finally, using the above argument with the commutator [Ln , L−n ], one can show that c, ˜c ≥ 0 in a unitary CFT. In fact, the only unitary CFT with c = 0 is the trivial one, Ln = 0. Zero-point energies The state–operator mapping gives a simple alternative derivation of the various normal ordering constants. In any CFT, we know that L0 |1 = 0, and this determines the additive normalization of L0 . In the X CFT, |1 is the ground state |0; 0 , so aX vanishes. In the bc theory, |1 is the excited state b−1 | ↓ , so the weight of | ↓ is −1 in agreement with the earlier result (2.7.21) for λ = 2.

2.9 More on states and operators

73

This also provides one physical interpretation of the central charge. In ˜ 0 = 0. The conformal a unitary CFT the ground state is |1 with L0 = L transformation (2.6.10) between the radial and time-translation generators then implies that E=−

c + ˜c 24

(2.9.17)

for the ground state. This is a Casimir energy, coming from the ﬁnite size of the system, and it depends only on the central charge. On dimensional grounds this energy is inverse to the size of the system, which is 2π here, so the general result would be E=−

π(c + ˜c) . 12

(2.9.18)

We have now given three honest ways of calculating normal ordering constants: from the Virasoro algebra as in eq. (2.7.8), by relating the two forms of normal ordering as in eq. (2.7.11), and from the state–operator mapping above. Nevertheless the idea of adding up zero-point energies is intuitive and a useful mnemonic, so we give a prescription that can be checked by one of the more honest methods: 1. Add the zero-point energies, 12 ω for each bosonic mode and − 12 ω for each fermionic (anticommuting) mode. 2. One encounters divergent sums of the form ∞ n=1 (n−θ), the θ arising when one considers nontrivial periodicity conditions. Deﬁne this to be ∞ n=1

(n − θ) =

1 1 − (2θ − 1)2 . 24 8

(2.9.19)

This is the value one obtains as in eq. (1.3.32) by regulating and discarding the quadratically divergent part. 3. The above gives the normal ordering constant for the w-frame generator T0 , eq. (2.6.8). For L0 we must add the nontensor correction 1 24 c. For the free boson, the modes are integer-valued so we get one-half of 1 . This is just oﬀset the sum (2.9.19) for θ = 0 after step 2, which is − 24 X by the correction in step 3, giving a = 0. For the ghosts we similarly get 2 26 24 − 24 = −1.

74

2 Conformal ﬁeld theory Exercises

2.1 Verify that 1 1 = ∂¯ = 2πδ 2 (z, ¯z ) ¯z z (a) by use of the divergence theorem (2.1.9); (b) by regulating the singularity and then taking a limit. ∂∂¯ ln |z|2 = ∂

2.2 Work out explicitly the expression (2.2.5) for the normal-ordered product of four X µ ﬁelds. Show that it is a harmonic function of each of the positions. 2.3 The expectation value of a product of exponential operators on the plane is n

iki ·X(zi ,¯zi )

:e

:

= iC X (2π)D δ D (

i=1

n i=1 ki )

n

|zij |α ki ·kj ,

i,j=1 i<j

with C X a constant. This can be obtained as a limit of the expectation value (6.2.17) on the sphere, which we will obtain by several methods in chapter 6. (a) Show that the leading behavior as one vertex operator approaches another is in agreement with the OPE (2.2.14). (b) As implied by eq. (2.2.13), the expectation value is |z12 |α k1 ·k2 times a function that is smooth as z1 → z2 . For n = 3, work out the explicit expansion of this smooth function in powers of z12 and ¯z12 . From the ratio of the large-order terms ﬁnd the radius of convergence. (c) Give a general proof that the free-ﬁeld OPE (2.2.4) is convergent inside the dashed circle in ﬁgure 2.1. 2.4 Extend the OPE (2.2.14) up to and including subleading terms of order z 2 , z¯z , and ¯z 2 . Show that the z, ¯z -dependence of each term is in agreement with the general form (2.4.20). Give also the symmetric form as in eq. (2.2.15) and discuss the properties of the coeﬃcients under interchange of the two operators. 2.5 Derive the classical Noether theorem in the form usually found in textbooks. That is, assume that

S [φ] =

dd σ L(φ(σ), ∂a φ(σ))

and ignore the variation of the measure. Invariance of the action implies that the variation of the Lagrangian density is a total derivative, δL = ∂a Ka under a symmetry transformation (2.3.1). Then the classical result, with

75

Exercises conventional string normalization, is

∂L −1 j = 2πi δφα − Ka ∂φα,a a

.

Derive the Lagrangian equations of motion and show that they imply that this current is conserved. Show that it agrees with eq. (2.3.4). 2.6 Consider the ﬂat Euclidean metric δab , in d dimensions. An inﬁnitesimal coordinate transformation δσ a = v a (σ) changes the metric by δgab = −∂a vb − ∂b va . Determine the most general v a (σ) that leaves the metric invariant up to a local rescaling. There are 12 (d + 1)(d + 2) independent solutions in d > 2, and an inﬁnite number in d = 2. [Although the result is given in many places in the literature, the author has not found a simple derivation.] 2.7 (a) By computing the relevant OPEs, conﬁrm the weights stated in eq. (2.4.17) and determine which operators are tensors. (b) Do this for the same operators in the linear dilaton theory. ¯ ν eik·X : ? What are the conditions on 2.8 What is the weight of fµν : ∂X µ ∂X fµν and kµ in order for it to be a tensor? 2.9 Derive the central charges for the linear dilaton, bc, and βγ CFTs by working out the T T OPEs. 2.10 Consider now a patch of world-sheet with boundary. For convenience suppose that the patch lies in the upper half-z-plane, with the real axis being the boundary. Show that expectation values of a normal-ordered ¯ µ (¯z ) :, although ﬁnite in the interior, diverge as operator, say : ∂X µ (z)∂X z approaches the boundary (represent the eﬀect of the boundary by an image charge). Deﬁne boundary normal ordering , which is the same as : : except that the contraction includes the image charge piece as well. Operators on the boundary are ﬁnite if they are boundary normal ordered. In a general CFT on a manifold with boundary, the interior operators and the boundary operators are independent. Label the bases Ai and Br respectively, and deﬁne the former by : : and the latter by . Each set has its own closed OPE: AA → A and BB → B. In addition, the sets are related: Ai , as it approaches the boundary, can be expanded in terms of the Br . Find the leading behaviors of

eik1 ·X(y1 ) eik2 ·X(y2 ) , ik·X(z,¯z )

:e

:,

y1 → y2 (y real) ,

Im(z) → 0 .

The identity (2.7.14) is useful for the latter. 2.11 Evaluate the central charge in the Virasoro algebra for X µ by calcu-

76

2 Conformal ﬁeld theory

lating Lm (L−m |0; 0 ) − L−m (Lm |0; 0 ) . 2.12 Use the OPE and the contour results (2.6.14) and (2.6.15) to derive the commutators (2.7.5) and (2.7.17). 2.13 (a) Show that : b(z)c(z ) : − ◦◦ b(z)c(z )◦◦ =

(z/z )1−λ − 1 z − z

by the method of eq. (2.7.11). (b) Use this to determine the ordering constant in N g , eq. (2.7.22). You also need the conformal transformation (2.8.14). (c) Show that one obtains the same value for N g by a heuristic treatment of the ordering similar to that in section 2.9. 2.14 (a) Determine the ordering constant in Lg0 , eq. (2.7.21), using part (a) of the previous exercise. (b) Determine it using the heuristic rules in section 2.9. 2.15 Work out the state–operator mapping for the open string X µ CFT. 2.16 Let O be a tensor ﬁeld of weight (h, ˜h). Evaluate the commutators [Lm , O(z, ¯z )] using the contour argument of ﬁgure 2.4. Show in particular that the commutator of a (1,1) tensor is a total derivative. 2.17 Extend the argument of eq. (2.9.14) to show that c ≥ 0.

3 The Polyakov path integral

We now begin a systematic study of string theory using the Polyakov path integral and conformal ﬁeld theory. After an introduction to the path integral picture, we discuss gauge ﬁxing, the Weyl anomaly, and vertex operators. We then generalize to strings propagating in curved spacetime. 3.1

Sums over world-sheets

The Feynman path integral is one way to represent a quantum theory, and it is a very natural method for describing interactions in string theory. In path integral quantization, amplitudes are given by summing over all possible histories interpolating between the initial and ﬁnal states. Each history is weighted by exp(iScl /¯h) ,

(3.1.1)

with Scl the classical action for the given history. Thus, one would deﬁne an amplitude in string theory by summing over all world-sheets connecting given initial and ﬁnal curves, as in ﬁgure 3.1(a) for the open string and ﬁgure 3.1(b) for the closed string. The analogous sum for the relativistic point particle produces the free propagator (exercise 5.1). One can imagine a number of ways in which strings might interact. One would be a contact interaction, an energy when two strings intersect. Another would be a long-range force mediated by some quantum ﬁeld. However, it is not possible to add such interactions to string theory in a way that is consistent with the symmetries; we will get some idea later of why this is so. Rather, the only interactions that are allowed are those that are already implicit in the sum over world-sheets. Consider, for example, the world-sheets shown in ﬁgure 3.2. Figure 3.2(a) looks like a quantum correction to the open string amplitude of ﬁgure 3.1(a), involving intermediate states with two open strings. Figure 3.2(b) has three external 77

78

3 The Polyakov path integral

(a)

(b)

Fig. 3.1. (a) An open string world-sheet with the topology of a strip. The heavier curves are the world-lines of string endpoints. (b) A closed string world-sheet with the topology of a cylinder.

(a)

(b)

Fig. 3.2. (a) Quantum correction to open string propagation. (b) Decay of one closed string into two. The dashed lines are time-slices.

closed strings and represents one string decaying into two. It turns out that this is the correct way to introduce interactions into string theory. We will see that these interactions produce a consistent theory, ﬁnite and unitary, which includes gravity. It is interesting to consider the process of ﬁgure 3.2(b) as seen at successive times as shown in ﬁgure 3.3. A closed string splits into two, or in the reverse process two join into one. This is the basic closed string interaction. In closed string theory, all particles are obtained as various states of excitation of the string, and all interactions (gauge, gravitational,

3.1 Sums over world-sheets

79

Fig. 3.3. Successive time-slices from ﬁgure 3.2(b). The arrows indicate the orientation of the string.

(a)

(b)

(c)

(d) Fig. 3.4. Processes involving open strings: (a) open ↔ open + open; (b) closed ↔ open; (c) open + open ↔ open + open; (d) open ↔ open + closed.

and Yukawa, for example) arise from the single process of ﬁgure 3.3. For theories with open strings there are several additional processes that can occur, as shown in ﬁgure 3.4. With a given slicing, the interactions of ﬁgures 3.3 and 3.4 appear to occur at a deﬁnite spacetime point, but this is an illusion. A diﬀerent slicing, in a boosted Lorentz frame, will put the apparent interaction at a diﬀerent point; there is no distinguished point in spacetime. The interaction arises only from the global topology of the world-sheet, while the local properties of the world-sheet are the same as they were in the free case. As discussed in the introduction, it is this smearing out of the interaction that cuts oﬀ the short-distance divergences of gravity.

80

3 The Polyakov path integral

(a)

(b)

Fig. 3.5. Oriented (a) and unoriented (b) contributions to the two-open-string amplitude.

There are actually several diﬀerent string theories, depending on which topologies we include in the sum over world-sheets. First note that the world-sheets we have drawn have two kinds of boundary, the source boundaries corresponding to the initial and ﬁnal string conﬁgurations, and the endpoint boundaries corresponding to the world-lines of the endpoints of open strings. Ignore for now the source boundaries; we will discuss the details of the sources later in this chapter. So in the following discussion ‘boundary’ means endpoint boundary. There are then four ways to deﬁne the sum over world-sheets, which correspond to the four free string theories listed at the end of section 1.4: 1. Closed oriented: All oriented world-sheets without boundary. 2. Closed unoriented: All world-sheets without boundary. 3. Closed plus open oriented: All oriented world-sheets with any number of boundaries. 4. Closed plus open unoriented: All world-sheets with any number of boundaries. There are two things that we should expand upon here. The ﬁrst is the connection between the inclusion of unoriented world-sheets and the Ω = +1 projection described in section 1.4. Figure 3.5 shows two worldsheets that are included in the unoriented theory with open strings. The unoriented world-sheet of ﬁgure 3.5(b) is equivalent to cutting the oriented surface of ﬁgure 3.5(a) along the dashed line, which we will parameterize

3.1 Sums over world-sheets

(a)

81

(b)

Fig. 3.6. (a) An annulus: the intermediate state is two open strings. (b) The same topology drawn as a cylinder: the intermediate state is one closed string.

by 0 ≤ σ ≤ π, and requiring the ﬁelds on the cut edges to satisfy µ µ Xupper (σ) = Xlower (π − σ) .

(3.1.2)

This in turn is equivalent to acting with the operator Ω on the open string state. Adding these two surfaces with appropriate weight then amounts to inserting the operator 12 (1 + Ω), which projects onto states of Ω = +1. Summing over all oriented and unoriented surfaces inserts projection operators on all intermediate states, reducing the spectrum to the Ω = +1 sector. Second, the list above does not include any theory with open strings but not closed. Let us explain in more detail why this is the case. Consider a world-sheet with the topology of an annulus, ﬁgure 3.6(a). For convenience we have drawn only the vacuum amplitude, but by attaching appropriate sources the same argument will also apply to scattering amplitudes. We see from the dashed line that this is a process with an intermediate state of two open strings, and so world-sheets of this topology must be present in any theory with open strings. In ﬁgure 3.6(b) we have drawn the same topology as a cylinder and cut it open on an intermediate state of a single closed string. Thus, even if we start with open strings only, the sum over world-sheets will necessarily include processes in which scattering of open strings produces closed strings. As we develop the amplitudes we will see how this works in detail. The need for closed strings in open string theory can also be seen in another way. Consider the interactions in ﬁgures 3.4(a) and (b), but time reversed: respectively two open strings joining into one, and an open to closed string transition. Near the interaction point these are the same, two endpoints coalescing. To have the ﬁrst interaction without the second would require some nonlocal constraint on the dynamics of the string; this would surely be inconsistent. One can make the same argument with ﬁgures 3.4(c) and (d), where the interaction is locally the reconnection of

82

3 The Polyakov path integral

a pair of strings. So if any open string interaction is allowed, then so is some process in which closed strings are produced from open strings. 3.2

The Polyakov path integral

We now begin to develop the sum over world-sheets, integrating over the ﬁelds of the Polyakov formalism. We make one change from chapter 1: the Minkowskian world-sheet metric γab is replaced with a Euclidean worldsheet metric gab (σ 1 , σ 2 ), of signature (+, +). The integral runs over all Euclidean metrics and over all embeddings X µ (σ 1 , σ 2 ) of the world-sheet in Minkowski spacetime:

[dX dg] exp(−S ) .

(3.2.1)

S = SX + λχ ,

(3.2.2)

The Euclidean action is

with

1 d2 σ g 1/2 g ab ∂a X µ ∂b Xµ , (3.2.3a) SX = 4πα M 1 1 χ= d2 σ g 1/2 R + ds k . (3.2.3b) 4π M 2π ∂M The geodesic curvature k was deﬁned in exercise 1.3; note that the boundary now is always spacelike. The advantage of the Euclidean path integral is that the integral over metrics is better deﬁned. The topologically nontrivial world-sheets we have been describing can have nonsingular Euclidean metrics but not Minkowskian ones. We will take the Euclidean theory (3.2.1) as our starting point, and we will show how to give it a precise deﬁnition and that it deﬁnes a consistent spacetime theory with the advertised properties. However, let us give a brief formal argument that it is equivalent to the Minkowski theory with which we began. We start again with the example of the point particle, where the path integral

[dη dX] exp

i 2

˙ µX ˙ µ − ηm2 dτ η −1 X

(3.2.4)

is oscillatory. The path integral over η and X µ is a product of ordinary integrals, and so we can deform contours just as in ordinary integration. If we take η(τ) → e−iθ η(τ) ,

X 0 (τ) → e−iθ X 0 (τ) ,

(3.2.5)

83

3.2 The Polyakov path integral

for inﬁnitesimal θ, all terms in the exponent acquire a negative real part and so this acts as a convergence factor. Now we can perform a contour rotation in ﬁeld space, all the way to η = −ie, X 0 = −iX D . The integral becomes

1 [de dX] exp − 2

−1

dτ e

D

˙ µX ˙ µ + em2 X

.

(3.2.6)

µ=1

This is the Euclidean point-particle analog of the path integral (3.2.1). We have just made a contour rotation, so the Euclidean path integral gives the same amplitude as the Minkowski one. The same procedure works for the Polyakov action if we write the metric in terms of a tetrad, γab = −e0a e0b + e1a e1b , and make the same rotation on e0a . This provides a formal justiﬁcation for the equivalence of the Minkowski and Euclidean path integrals.1 It has been shown by explicit calculation that they deﬁne the same amplitudes, respectively in the light-cone and conformal gauges. We have in the action (3.2.3a) left the rotation of X 0 implicit, as in the previous chapter, so as to emphasize the Minkowski nature of spacetime. Written in terms of X 0 , the equations of motion and the OPE are covariant, with metric η µν . We have noted that χ is locally a total derivative in two dimensions and therefore depends only on the topology of the world-sheet — it is the Euler number of the world-sheet. Thus, the e−λχ factor in the path integral aﬀects only the relative weighting of diﬀerent topologies in the sum over world-sheets. In particular, if one adds an extra strip to a world-sheet, as we have done in going from ﬁgure 3.1(a) to ﬁgure 3.2(a), the Euler number decreases by one and the path integral is weighted by an extra factor of eλ . Since adding a strip corresponds to emitting and reabsorbing a virtual open string, the amplitude for emitting an open string from any process is proportional to eλ/2 . Adding a handle to any world-sheet reduces the Euler number by 2 and adds a factor e2λ . Since this corresponds to emitting and reabsorbing a closed string, the amplitude for emitting a closed string is proportional to eλ . The Euler term in the action thus controls the coupling constants in the string theory, with go2 ∼ gc ∼ eλ .

(3.2.7)

By the way, λ might seem to be a free parameter in the theory, contradicting the statement in the introduction that there are no such parameters 1

In more than two dimensions, things are not so simple because the Hilbert action behaves in a more complicated way under the rotation. No simple rotation damps the path integral. In particular, the meaning of the Euclidean path integral for four-dimensional gravity is very uncertain.

84

3 The Polyakov path integral

in string theory. This is an important point, and we will resolve it in section 3.7. As an aside, the counting of couplings extends in a simple way to world-sheets with stringy sources. The sources for closed strings are closed boundary loops, while the open string sources are boundary segments. For convenience we will require the open string source boundaries to meet the endpoint boundaries at right angles in the world-sheet metric. We have to be careful because the boundary curvature k diverges at the corners: the Euler number is 1 (3.2.8) χ = ˜χ + nc , 4 where ˜χ includes only the integral over the smooth segments of boundary and nc is the number of corners. The correct weight for the path integral is exp(−λ˜χ) = exp(−λχ + λnc /4) .

(3.2.9)

This follows from unitarity: the λ-dependence must be unchanged if we cut through a world-sheet, leaving fewer internal lines and more external sources. This will be the case for the weight (3.2.9) because the surface integral in ˜ χ cancels on the cut edges. Adding a closed string source amounts to cutting a hole in the world-sheet and decreases χ by 1. Adding an open string source leaves χ unchanged and increases nc by 2. Thus we reach the same conclusion (3.2.7) about the amplitude for emission of closed or open strings. 3.3

Gauge ﬁxing

The path integral (3.2.1) is not quite right. It contains an enormous overcounting, because conﬁgurations (X, g) and (X , g ) that are related to one another by the local diﬀ×Weyl symmetry represent the same physical conﬁguration. We need, in eﬀect, to divide by the volume of this local symmetry group,

[dX dg] exp(−S ) ≡ Z . Vdiﬀ×Weyl

(3.3.1)

We will carry this out by gauge-ﬁxing, integrating over a slice that cuts through each gauge equivalence class once and obtaining the correct measure on the slice by the Faddeev–Popov method. In chapter 1 we imposed light-cone gauge. This is useful for some purposes but hides some of the symmetry of the theory, so we now make a diﬀerent choice. Note that the metric, being symmetric, has three independent components, and that there are three gauge functions, the

3.3 Gauge ﬁxing

85

two coordinates and the local scale of the metric. Thus there is just enough gauge freedom to eliminate the integration over the metric, ﬁxing it at some speciﬁc functional form which we will call the ﬁducial metric, gab (σ) → gˆ ab (σ) .

(3.3.2)

A simple choice is the ﬂat or unit gauge metric gˆ ab (σ) = δab .

(3.3.3)

One sometimes wishes to consider the eﬀect of the diﬀ group alone. In this case, one can bring an arbitrary metric to within a Weyl transformation of the unit form. This is the conformal gauge, gˆ ab (σ) = exp[2ω(σ)]δab .

(3.3.4)

Let us see explicitly that this counting of ﬁelds and gauge symmetries works — that any metric can be brought to the ﬂat form at least locally, in a given neighborhood on the world-sheet. First, make a Weyl transformation to set to zero the Ricci scalar built from gab . The Weyl transformation of the Ricci scalar is g 1/2 R = g 1/2 (R − 2∇2 ω) .

(3.3.5)

To set R = 0 requires solving 2∇2 ω = R as an equation for ω. This is always possible, at least locally, by a standard argument from electrostatics. Now, in two dimensions the vanishing of the Ricci scalar implies the vanishing of the entire Riemann tensor, because the symmetries of the Riemann tensor imply that 1 (3.3.6) Rabcd = (gac gbd − gad gbc )R . 2 So the metric is ﬂat and coordinate-equivalent to the unit metric (3.3.3). Actually, locally there is a little bit of extra gauge freedom. There are diﬀ×Weyl transformations that leave the metric in unit gauge and so are not ﬁxed by the choice of metric. It is convenient here to introduce a complex coordinate z = σ 1 + iσ 2 as in chapter 2, where the ﬂat metric is ds2 = dzd¯z . Consider a coordinate transformation such that z is a holomorphic function of z, z ≡ σ 1 + iσ 2 = f(z) ,

(3.3.7)

combined with a Weyl transformation. The new metric is ds2 = exp(2ω)|∂z f|−2 dz d¯z .

(3.3.8)

ω = ln |∂z f|

(3.3.9)

Then for

86

3 The Polyakov path integral

the metric is invariant. This extra gauge freedom does not conﬂict with our earlier counting (3 = 3) because the mismatch, the set of holomorphic reparameterizations, is of measure zero compared to all reparameterizations. When we consider the full world-sheet, most or all of this extra freedom will be eliminated by the boundary conditions. There are two issues here. First, in the discussion of Noether’s theorem and the Ward identities, we were careful to emphasize below eq. (2.3.10) that these results depend only on the symmetry being deﬁned locally on the world-sheet. The transformation (3.3.7), (3.3.9) will therefore give rise to a conserved current and to Ward identities. This is precisely the conformal invariance studied in chapter 2. We see that it arises as the subgroup of diﬀ×Weyl that leaves invariant the unit metric. The second issue is what happens to our counting of metric and gauge degrees of freedom when we do consider the full world-sheet. We will take this up in chapter 5, where we will see that globally there is a small mismatch between the two, a ﬁnite number of parameters. Until then, we will be concerned only with local properties of the world-sheet theory. The Faddeev–Popov determinant After ﬁxing the metric, the functional integral runs along a slice parameterized by X µ alone. In order to obtain the correct measure, we follow the Faddeev–Popov procedure. The steps are the same as were used to obtain the gauge-ﬁxed measure in Yang-Mills theory, but we will give a self-contained discussion. The idea, illustrated in ﬁgure 3.7, is to separate the path integral into an integral over the gauge group times an integral along the gauge slice, and to divide out the former. The Faddeev–Popov determinant is the Jacobian for this change of variables. Let ζ be shorthand for a combined coordinate and Weyl transformation, ∂σ c ∂σ d gcd (σ) . (3.3.10) ∂σ a ∂σ b Following a standard route, we deﬁne the Faddeev–Popov measure ∆FP by ζ : g → gζ ,

ζ gab (σ ) = exp[2ω(σ)]

1 = ∆FP (g)

[dζ] δ(g − gˆ ζ ) ,

(3.3.11)

where again gˆ ab is the ﬁducial metric. In (3.3.11), [dζ] is a gauge-invariant measure on the diﬀ×Weyl group. We will not need the explicit form of this measure; we will discuss the question of its existence in the next section. ζ (σ) The delta function is actually a delta functional, requiring gab (σ) = gˆ ab at every point.

87

3.3 Gauge ﬁxing

field space gauge slice

gauge orbits Fig. 3.7. Schematic picture of the Faddeev–Popov procedure. The gauge orbits are families of gauge-equivalent conﬁgurations. Integrating over the whole ﬁeld space and dividing by the orbit volume is equivalent to integrating over a slice that intersects each orbit once, with appropriate Jacobian.

Inserting (3.3.11) into the functional integral (3.3.1), the latter becomes

ˆ = Z[g]

[dζ dX dg] ∆FP (g)δ(g − gˆ ζ ) exp(−S [X, g]) . Vdiﬀ×Weyl

(3.3.12)

For later reference, we denote explicitly the dependence of Z on the choice of ﬁducial metric. Carry out the integration over gab , and also rename the dummy variable X → X ζ , to obtain

ˆ = Z[g]

[dζ dX ζ ] ∆FP (gˆ ζ ) exp(−S [X ζ , gˆ ζ ]) . Vdiﬀ×Weyl

(3.3.13)

Now use the gauge invariance2 of ∆FP , of the functional integral measure [dX], and of the action to obtain

ˆ = Z[g]

[dζ dX] ˆ exp(−S [X, g]) ˆ . ∆FP (g) Vdiﬀ×Weyl

(3.3.14)

Finally, nothing in the integrand depends on ζ, so the integral over ζ just produces the volume of the gauge group and cancels the denominator, 2

To show this invariance:

∆FP (g ζ )−1 =

=

[dζ ] δ(g ζ − gˆ ζ ) =

[dζ ] δ(g − gˆ ζ

−1 ·ζ

)

[dζ ] δ(g − gˆ ζ ) = ∆FP (g)−1 ,

where ζ = ζ −1 · ζ . In the second equality we have used the gauge invariance of the delta function, and in the third the invariance of the measure.

88

3 The Polyakov path integral

leaving

ˆ = Z[g]

ˆ exp(−S [X, g]) ˆ . [dX] ∆FP (g)

(3.3.15)

ˆ is the correct measure on the slice. Thus, ∆FP (g) To evaluate the expression (3.3.11) for the Faddeev–Popov measure, let us pretend that the gauge choice precisely ﬁxes the gauge symmetry, so that for exactly one value of ζ the delta function δ(g − gˆ ζ ) is nonzero; for δ(gˆ − gˆ ζ ) this would be the identity. As noted above, there is a small global mismatch, but we will deal with this when the time comes — it does not aﬀect the local properties of the world-sheet. For ζ near the identity, we can expand δgab = 2δω gab − ∇a δσb − ∇b δσa = (2δω − ∇c δσ c )gab − 2(P1 δσ)ab .

(3.3.16)

We have deﬁned a diﬀerential operator P1 that takes vectors into traceless symmetric 2-tensors, 1 (P1 δσ)ab = (∇a δσb + ∇b δσa − gab ∇c δσ c ) . 2

(3.3.17)

Near the identity, the inverse determinant (3.3.11) becomes ˆ −1 ∆FP (g)

=

=

!

[dδω dβ dδσ] exp 2πi

=

ˆ [dδω dδσ] δ −(2δω − ∇·δσ) gˆ + 2Pˆ 1 δσ !

[dβ dδσ] exp 4πi

2

d σ gˆ

1/2 ab

β

ˆ −(2δω − ∇·δσ) gˆ + 2Pˆ 1 δσ

d2 σ gˆ 1/2 β ab (Pˆ 1 δσ)ab

" ab

"

.

(3.3.18)

A hat on a diﬀerential operator indicates that it contains the ﬁducial ˆ In the second equality we have used the integral representation metric g. for the functional delta function, introducing the symmetric tensor ﬁeld β ab . In the third equality we have integrated over δω, which produces a delta functional forcing β ab to be traceless; the functional integral [dβ ] is then over traceless symmetric tensors. We now have a representation ˆ −1 as a functional integral over a vector ﬁeld δσ a and over a for ∆FP (g) traceless symmetric tensor β ab . As discussed in the appendix, we can invert this path integral by replacing each bosonic ﬁeld with a corresponding Grassmann ghost ﬁeld. δσ a → ca , βab → bab ,

(3.3.19a) (3.3.19b)

3.3 Gauge ﬁxing

89

with bab , like β ab , being traceless. Thus,

ˆ = ∆FP (g)

[db dc] exp(−Sg ) ,

(3.3.20)

where the ghost action Sg , with a convenient normalization for the ﬁelds, is 1 ˆ a cb = 1 d2 σ gˆ 1/2 bab ∇ d2 σ gˆ 1/2 bab (Pˆ 1 c)ab . (3.3.21) Sg = 2π 2π Locally on the world-sheet, the Polyakov path integral is now

ˆ = Z[g]

[dX db dc] exp(−SX − Sg ) .

(3.3.22)

The action is quadratic in the ﬁelds, so we can evaluate the path integral in terms of determinants as discussed in the appendix. This step will be carried out in more detail in the chapter 5; the result here is ˆ 2 )−D/2 det Pˆ 1 , ˆ = (det ∇ Z[g] (3.3.23) the ﬁrst determinant coming from the X integration and the second from the ghost integration. In conformal gauge, gˆ ab (σ) = exp[2ω(σ)]δab , the ghost action (3.3.21) is 1 Sg = d2 z bzz ∇¯z cz + b¯z¯z ∇z c¯z 2π 1 = d2 z bzz ∂¯z cz + b¯z¯z ∂z c¯z . (3.3.24) 2π Notice that ω(σ) does not appear in the ﬁnal form. This is because the covariant ¯z -derivative of a tensor that has only z indices reduces to the simple derivative, and vice versa, as the reader can check by working out the connection tensor. We have positioned the indices, lower on b and upper on c, to take advantage of this. Since the action (3.3.24) is Weyl-invariant, it follows that bab and ca must be neutral under the Weyl transformation (in contrast to bab and ca , for example, which contain additional factors of the metric). Since the conformal transformation is a combination of a coordinate transformation (3.3.7) and a Weyl transformation (3.3.9), we can immediately read oﬀ the conformal transformations of bab and ca from the tensor indices: this is a bc CFT with (hb , hc ) = (2, −1) and a ˜b˜c hc ) = (2, −1). CFT with (˜hb , ˜ The derivation that we have given of the gauge-ﬁxed path integral is sometimes referred to as the ‘heuristic’ derivation. The problem is that the starting point, the Polyakov functional integral (3.3.1), is ill deﬁned due to the enormous gauge overcounting. The gauge-ﬁxed functional integral (3.3.22), on the other hand, is quite well deﬁned, and can be calculated explicitly. For practical purposes we should perhaps regard the gaugeﬁxed form as our true starting point, with the gauge-invariant one giving

90

3 The Polyakov path integral

an intuitive interpretation. In the gauge-ﬁxed form, the essential content of the gauge symmetry is still present as BRST invariance, which we will take up in the next chapter. In the open string we need also to consider a patch containing a segment of the boundary. It is convenient to regard functional integrals as being over ﬁelds on a ﬁxed coordinate region, so the diﬀ invariance is limited to coordinate changes that take the boundary into itself. The variation δσ a therefore has vanishing normal component, na δσ a = 0 .

(3.3.25)

This boundary condition is inherited by the corresponding ghost ﬁeld, na ca = 0 ,

(3.3.26)

so ca is proportional to the tangent vector ta . The equations of motion then provide a boundary condition on bab . They have a surface term

∂M

ds na bab δcb = 0 .

(3.3.27)

With the boundary condition on ca , this implies na tb bab = 0 .

(3.3.28)

These are the boundary conditions that we used in section 2.7. 3.4

The Weyl anomaly

A key feature of string theory is that it is not consistent in all spacetime backgrounds, but only in those satisfying certain conditions. For the bosonic string theory in ﬂat spacetime, the condition is D = 26. This was ﬁrst discovered as a pathology of the scattering amplitudes. In the light-cone analysis, it arises as a loss of Lorentz invariance, as we brieﬂy and rather heuristically explained in section 1.3. The underlying source of the restriction is an anomaly in the local world-sheet symmetries. In our present formalism, the anomaly is in the Weyl symmetry: T aa vanishes classically, but not in the quantum theory. The reader who has studied quantum ﬁeld theory has already encountered a related anomaly, although it was likely not described in the same language. A global Weyl transformation, ω(σ) = constant, is equivalent to an overall rescaling of lengths. A number of familiar ﬁeld theories in four dimensions are classically invariant under such a rescaling. These include massless scalar ﬁeld theory with a φ4 interaction, and non-Abelian gauge theory. The scale invariance is evident because the Lagrangian contains no dimensionful parameters — the coupling constant in each case is dimensionless. However, we know that due to divergences in the quantum

3.4 The Weyl anomaly

91

theory there is a nonzero renormalization group beta function in each theory, implying that the eﬀective coupling constant is in fact a function of the length scale. Correspondingly, the trace of the energy-momentum tensor vanishes in the classical theory but is nonzero and proportional to the beta function when quantum eﬀects are taken into account. In the previous section, we ignored the possibility of anomalies. We need to check this: is the gauge-ﬁxed path integral Z[g] really independent of the choice of ﬁducial metric, Z[g ζ ] = Z[g] ?

(3.4.1)

In section 4.2 we will extend this to more general changes of gauge. For convenience, we omit the hat on g henceforth. We will also be interested in path integrals with additional insertions,

. . . g ≡

[dX db dc] exp(−S [X, b, c, g]) . . . .

(3.4.2)

Then we also require

. . . g ζ = . . . g .

(3.4.3)

We are not at the present time interested in the details of the insertions themselves — that will be the subject of the next two sections — so we will restrict attention to gauge transformations ζ vanishing near the positions of the ﬁelds in ‘. . .’. That is, we are asking that Weyl invariance hold as an operator equation. It is easy to preserve the diﬀ and Poincar´e invariances in the quantum theory. For example, one may deﬁne the gauge-ﬁxed path integral using a Pauli–Villars regulator as is done for the harmonic oscillator in section A.1, dividing by the path integral for a massive regulator ﬁeld. The massive regulator ﬁeld can be coupled to the metric in a diﬀ- and Poincar´e-invariant way. However, the diﬀ- and Poincar´e-invariant mass term µ2 d2 σ g 1/2 Y µ Yµ for a regulator ﬁeld Y µ is not Weyl-invariant. We must therefore check Weyl invariance by hand: does the invariance (3.4.3) hold for Weyl transformations? Any Weyl transformation can be obtained as the result of repeated inﬁnitesimal transformations, so it is suﬃcient to study the latter. The energy-momentum tensor T ab is the inﬁnitesimal variation of the path integral with respect to the metric, 1 δ . . . g = − d2 σ g(σ)1/2 δgab (σ) T ab (σ) . . . . (3.4.4) g 4π Classically, T ab comes entirely from the variation of the action, and this coincides with the Noether deﬁnition 4π δ classical S . (3.4.5) T ab (σ) = 1/2 g(σ) δgab (σ)

92

3 The Polyakov path integral

It also reduces to the earlier deﬁnition in the limit of a ﬂat world-sheet. If one takes δgab to have the form of a coordinate transformation, the diﬀ invariance of the path integral implies the conservation of T ab . In the following analysis, we will at ﬁrst neglect possible boundary terms. For a Weyl transformation, the deﬁnition (3.4.4) of T ab means that

1 d2 σ g(σ)1/2 δω(σ) T aa (σ) . . . g , (3.4.6) 2π so Weyl invariance with the general insertions ‘. . .’ can be phrased as the operator statement that the energy-momentum tensor is traceless: δW . . . g = −

?

T aa = 0 .

(3.4.7)

The classical action is Weyl-invariant, but in the quantum theory the trace might be nonzero because we have not managed to ﬁnd a fully gauge-invariant regulator. The trace must be diﬀ- and Poincar´e-invariant, because we have preserved these symmetries, and it must vanish in the ﬂat case because we know from the previous chapter that that theory is conformally invariant. This leaves only one possibility, T aa = a1 R

(3.4.8)

for some constant a1 , with R again the world-sheet Ricci scalar. Terms with more than two derivatives are forbidden for dimensional reasons. In units of world-sheet length, gab and X µ are both dimensionless, so the constant a1 in eq. (3.4.8) is dimensionless as well. Terms with more derivatives would have a coeﬃcient with positive powers of the cutoﬀ length used to deﬁne the path integral, and so vanish in the limit where the cutoﬀ is taken to zero. Calculation of the Weyl anomaly The possible obstruction to a gauge-invariant theory has been reduced to the single number a1 . In fact, this number is proportional to something we have already calculated, the central charge c of the CFT on a ﬂat world-sheet. Both a1 and c are related to the two-point function of the energy-momentum tensor, though to diﬀerent components. To get the precise relation between them we will need to use the diﬀ invariance, as follows. In conformal gauge, a1 (3.4.9) Tz¯z = gz¯z R . 2 Taking the covariant derivative, a1 a1 ∇¯z T¯z z = ∇¯z (gz¯z R) = ∂z R , (3.4.10) 2 2

93

3.4 The Weyl anomaly

where we have used the fact that the metric is covariantly constant. Then by conservation of Tab , a1 (3.4.11) ∇z Tzz = −∇¯z T¯z z = − ∂z R . 2 To ﬁx a1 we compare the Weyl transformations on the left and right. The Weyl transformation (3.3.5) of the right is a1 ∂z ∇2 δω ≈ 4a1 ∂z2 ∂¯z δω ,

(3.4.12)

where we have expanded around a ﬂat world-sheet. To get the Weyl transformation of Tzz we use ﬁrst the conformal transformation (2.4.24), c (3.4.13) −1 δTzz (z) = − ∂z3 v z (z) − 2∂z v z (z)Tzz (z) − v z (z)∂z Tzz (z) . 12 From the discussion in section 3.3, this conformal transformation consists of a coordinate transformation δz = v plus a Weyl transformation 2δω = ∂v + (∂v)∗ . The last two terms in the variation are the coordinate transformation of the tensor, so the Weyl transformation, to leading order around ﬂat space, is c (3.4.14) δW Tzz = − ∂z2 δω . 6 Acting with ∂z = 2∂¯z and comparing with the transformation (3.4.12) gives c c = −12a1 , T aa = − R . (3.4.15) 12 Let us derive this again by a slightly longer route, along which we obtain a useful intermediate result. In conformal gauge, R = −2 exp(−2ω)∂a ∂a ω , ∇2 = exp(−2ω)∂a ∂a .

(3.4.16a) (3.4.16b)

By the contraction with two lowered indices we mean δ ab ∂a ∂b . The Weyl variation (3.4.6) of Z[g] becomes

a1 (3.4.17) Z[exp(2ω)δ] d2 σ δω∂a ∂a ω , π where ‘exp(2ω)δ’ denotes conformal gauge. This integrates immediately to δW Z[exp(2ω)δ] =

Z[exp(2ω)δ] = Z[δ] exp −

a1 2π

d2 σ ∂a ω∂a ω

.

(3.4.18)

Since every metric is diﬀ-equivalent to a conformal metric, eq. (3.4.18) actually determines the complete metric-dependence of Z[g]. We need to ﬁnd the diﬀ-invariant expression that reduces to eq. (3.4.18) in conformal gauge. Using the relations (3.4.16), one veriﬁes that the desired expression

94

3 The Polyakov path integral

is

a1 Z[g] = Z[δ] exp 8π

2

d σ

2 1/2

d σg

R(σ)G(σ, σ )g

1/2

R(σ ) ,

(3.4.19)

where g(σ)1/2 ∇2 G(σ, σ ) = δ 2 (σ − σ )

(3.4.20)

deﬁnes the scalar Green’s function. This is an interesting result: the path integral is completely determined by the anomaly equation. Now expand Z[g] around a ﬂat background, gab = δab + hab , and keep terms of second order in h¯z¯z . The Ricci scalar is 4∂z2 h¯z¯z to ﬁrst order in h¯z¯z , so (ignoring contact terms, where z = z )

Z[δ + h] a1 ln ≈ 2 d2 z d2 z (∂z2 ln |z − z |2 )h¯z¯z (z, ¯z )∂z2 h¯z¯z (z , ¯z ) Z[δ] 8π 3a1 h¯z¯z (z, ¯z )h¯z¯z (z , ¯z ) = − 2 d2 z d2 z . (3.4.21) 4π (z − z )4 We can also calculate this by using second order perturbation theory in the metric, which by eq. (3.4.4) gives 1 Z[δ + h] ≈ 2 ln Z[δ] 8π

2

d z

d2 z h¯z¯z (z, ¯z )h¯z¯z (z , ¯z ) Tzz (z)Tzz (z )

δ

.

(3.4.22) Now use the standard T T OPE (2.4.25). All terms except the ﬁrst involve operators of nonzero spin and so vanish by rotational invariance, leaving

Tzz (z)Tzz (z ) δ = 12 c(z − z )−4 . Comparison with the result (3.4.21) from the Weyl anomaly gives again eq. (3.4.15). Discussion The world-sheet theory consists of the X µ , with central charge D, and the ghosts, whose central charge (2.5.12) is −26 for λ = 2, giving c = cX + cg = D − 26 .

(3.4.23)

The theory is Weyl-invariant only for D = 26, the same as the condition for Lorentz invariance found in chapter 1. When the Weyl anomaly is nonvanishing, diﬀerent gauge choices are inequivalent, and as with anomalous gauge theories any choice leads to a pathology such as loss of covariance or loss of unitarity. In light-cone gauge, for example, choosing diﬀerent spacetime axes requires a change of gauge, so the underlying Weyl anomaly is translated into a Lorentz anomaly. There is another thing that one might try: ignore the Weyl invariance and treat only the diﬀ invariance as a gauge symmetry. The metric would then have one real degree of freedom ω(σ), to be integrated over rather

3.4 The Weyl anomaly

95

than gauge-ﬁxed. This is particularly plausible because the quantum eﬀects in eq. (3.4.18) have generated an action for ω which looks just like that for one of the X µ ; for D < 26 the sign is that of a spacelike dimension. However, the physics is a little exotic because the diﬀ transformation (and therefore the energy-momentum tensor) of ω is diﬀerent from that of the X µ . In fact the theory is the linear dilaton CFT. So there is no (D + 1)-dimensional Lorentz invariance and this theory is not useful for our immediate purpose, which is to understand physics in our part of the universe. It is an interesting model, however, and we will return to this point brieﬂy in section 9.9. We have found above that a ﬂat world-sheet CFT can be coupled to a curved metric in a Weyl-invariant way if and only if the total central charge c is zero. There is a paradox here, however, because the same argument applies to the antiholomorphic central charge ˜c, and we have seen from the example of the bc CFT that c need not equal ˜c. The point is that we have implicitly assumed world-sheet diﬀ invariance in eqs. (3.4.11) and (3.4.19). If c = ˜c there is no diﬀ-invariant expression that agrees with both the O(h2zz ) and O(h¯2z¯z ) calculations. The CFT then cannot be coupled to a curved metric in a diﬀ-invariant way; there is a diﬀ or gravitational anomaly. The paradox shows that c = ˜c is necessary for a CFT to be free of gravitational anomalies, and it can also be shown that this is suﬃcient. Inclusion of boundaries Let us ﬁrst return to eq. (3.4.8) and consider the more general possibility T aa = a1 R + a2 .

(3.4.24)

This is allowed by diﬀ invariance but would be nonvanishing in the ﬂat limit. Earlier we set a2 to zero by the choice (2.3.15b) of Tab , explained in the footnote below that equation. Equivalently, the existence of more than one conserved Tab on the ﬂat world-sheet means that there is some freedom in how we couple to a curved metric. In particular, we can add to the action

Sct = b

d2 σ g 1/2 ,

(3.4.25)

which is a two-dimensional cosmological constant. This is not Weylinvariant. It adds 2πbgab to Tab and so the trace (3.4.24) becomes T aa = a1 R + (a2 + 4πb) .

(3.4.26)

We can therefore make the second term vanish by setting b = −a2 /4π; this is what we have done implicitly in eq. (2.3.15b). The separate values of a2 and b depend on deﬁnitions and are uninteresting.

96

3 The Polyakov path integral

No such local counterterm can remove a1 . For example, d2 σ g 1/2 R is Weyl-invariant, as we have already noted. So if a1 is nonzero, there is a real anomaly in the Weyl symmetry. We now consider the eﬀects of boundaries. Including boundary terms, the most general possible variation is3 1 δW ln . . . g = − d2 σ g 1/2 (a1 R + a2 )δω 2π M 1 − ds (a3 + a4 k + a5 na ∂a )δω , (3.4.27) 2π ∂M where again k = −ta nb ∇a tb is the geodesic curvature of the boundary. The possible counterterms are

Sct =

M

d2 σ g 1/2 b1 +

∂M

ds (b2 + b3 k) .

(3.4.28)

The b3 term is in addition to the geodesic curvature term in the Euler number, in eq. (3.2.3b). The Weyl variation of Sct , being careful to include the dependence of ds and of the unit vectors t and n on the metric, is

δW Sct = 2

M

d2 σ g 1/2 b1 δω +

∂M

ds (b2 + b3 na ∂a )δω .

(3.4.29)

We see that b1 , b2 , and b3 can be chosen to set a2 , a3 , and a5 to zero, leaving a1 and a4 as potential anomalies. There is one more tool at our disposal, the Wess–Zumino consistency condition. Take the second functional variation δW1 (δW2 ln . . . g ) a1 a4 2 1/2 2 = d σ g δω2 ∇ δω1 − ds δω2 na ∂a δω1 π M 2π ∂M a1 2a1 − a4 =− d2 σ g 1/2 ∂a δω2 ∂a δω1 + ds na δω2 ∂a δω1 . π M 2π ∂M (3.4.30) This is a second (functional) derivative and so by deﬁnition must be symmetric in δω1 and δω2 . The ﬁrst term is symmetric but the second one is not, and so a4 = 2a1 . It follows that boundaries, with open string boundary conditions, give rise to no new Weyl anomalies. Weyl invariance holds in the quantum theory if and only if a1 = −c/12 vanishes. This is determined by the local physics in the interior of the world-sheet. The Wess–Zumino consistency condition is a very powerful restriction on the form of possible anomalies. A second application enables us to show that the central charge is a constant. This followed above from 3

We require all terms to be independent of the ﬁelds X µ , as can be shown to hold in free ﬁeld theory. When we consider the interacting case in section 3.7, the Weyl anomaly will take a more general form.

3.5 Scattering amplitudes

97

Lorentz invariance and dimensional analysis, but in more general CFTs one might imagine that C(σ) R(σ) , (3.4.31) 12 where C(σ) is some local operator. In particular, the trace would still vanish on a ﬂat world-sheet. Repeating the calculation (3.4.30) with −C(σ)/12 in place of a1 , there is an additional asymmetric term proportional to ∂a C(σ) from the integration by parts. The consistency condition thus implies that expectation values involving C(σ) are position-independent, implying that C(σ) itself is just a numerical constant c. T aa (σ) = −

3.5

Scattering amplitudes

The idea of a sum over all world-sheets bounded by given initial and ﬁnal curves seems like a natural one. However, it is diﬃcult to deﬁne this in a way that is consistent with the local world-sheet symmetries, and the resulting amplitudes are rather complicated. There is one special case where the amplitudes simplify. This is the limit where the string sources are taken to inﬁnity. This corresponds to a scattering amplitude, an S-matrix element, with the incoming and outgoing strings speciﬁed. For most of this book we will conﬁne ourselves to the S-matrix and similar ‘on-shell’ questions. This is not completely satisfying, because one would like to be able to discuss the state of the system at ﬁnite times, not just asymptotic processes. For now we focus on the S-matrix because it is well understood, while it is not clear what is the correct way to think about string theory oﬀ-shell. We will return to this issue brieﬂy in the next section, and again in chapter 9. So we are now considering a process like that shown in ﬁgure 3.8(a), where the sources are pulled oﬀ to inﬁnity. We will give a heuristic argument for how these sources should be represented in the path integral. Away from the scattering process the strings propagate freely. Each of the incoming and outgoing legs is a long cylinder, which can be described with a complex coordinate w, − 2πt ≤ Im w ≤ 0 ,

w∼ = w + 2π .

(3.5.1)

The lower end of the cylinder, Im w = −2πt, is the end with the external source, the upper end ﬁts onto the rest of the world-sheet, and the circumference is the periodic Re w direction. The limit corresponding to the scattering process is t → ∞. It may seem that we are confusing a long distance in spacetime with a long cylinder in the world-sheet coordinate, but this will turn out to be correct: we will see later that propagation

98

3 The Polyakov path integral

(a)

(b)

Fig. 3.8. (a) Scattering of four closed strings, with sources approaching X 0 = ±∞. (b) Conformally equivalent picture of the four-closed-string scattering: a sphere with small holes.

over long spacetime distances comes precisely from world-sheets where a cylinder is growing long in the sense above. As we know from chapter 2, the cylinder has a conformally equivalent description in terms of the coordinate z, z = exp(−iw) ,

exp(−2πt) ≤ |z| ≤ 1 .

(3.5.2)

In this picture, the long cylinder is mapped into the unit disk, where the external string state is a tiny circle in the center. In the geometry of the internal metric, the process of ﬁgure 3.8(a) now looks like ﬁgure 3.8(b). In the limit t → ∞, the tiny circles shrink to points and the world-sheet reduces to a sphere with a point-like insertion for each external state. The same idea holds for external open string states. The long strips in the process of ﬁgure 3.9(a) can be described as the region − 2πt ≤ Im w ≤ 0 ,

0 ≤ Re w ≤ π ,

(3.5.3)

where Im w = −2πt is the source and Re w = 0, π are endpoint boundaries. Under z = − exp(−iw), this maps into the intersection of the unit disk with the upper half-plane, with a tiny semi-circle cut out at the origin, exp(−2πt) ≤ |z| ≤ 1 ,

Im z ≥ 0 .

(3.5.4)

99

3.5 Scattering amplitudes

(a)

(b)

Fig. 3.9. (a) Scattering of four open strings, with sources approaching X 0 = ±∞. (b) Conformally equivalent picture: a disk with small dents.

The scattering process now looks like ﬁgure 3.9(b). In the limit t → ∞, the sources shrink to points located on the (endpoint) boundary. This is the state–operator mapping, which we have already seen. Each string source becomes a local disturbance on the world-sheet. To a given incoming or outgoing string, with D-momentum k µ and internal state j, there corresponds a local vertex operator Vj (k) determined by the limiting process. Incoming and outgoing states are distinguished by the sign of k 0 ; for an incoming state k µ = (E, k) and for an outgoing state k µ = −(E, k). Figures 3.8 and 3.9 depict only the lowest order amplitudes, but the construction is clearly general: we may restrict attention to compact world-sheets, with no tubes going oﬀ to inﬁnity but with a pointlike insertion in the interior for each external closed string and a pointlike insertion on the boundary for each external open string. An n-particle S-matrix element is then given by Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

n [dX dg] exp(−SX − λχ) d2 σi g(σi )1/2 Vji (ki , σi ) . Vdiﬀ×Weyl i=1

(3.5.5)

To make the vertex operator insertions diﬀ-invariant, we have integrated

100

(a)

3 The Polyakov path integral

(b)

(c)

Fig. 3.10. Compact connected closed oriented surfaces of genus (a) 0, (b) 1, (c) 2.

them over the world-sheet. In the remainder of volume one, we will see that our guess is correct and that this does indeed deﬁne a sensible string S-matrix. Depending on which of the four types of string theories we are considering, the sum over topologies may include unoriented world-sheets, and/or world-sheets with boundary. In general, the sum over topologies is not restricted to connected world-sheets; the total process may involve two or more widely separated sets of particles scattering independently. It is convenient to focus on the connected S-matrix, given by restricting the sum to connected surfaces. To obtain the connected S-matrix, then, we must sum over all compact, connected topologies for the world-sheet. In two dimensions, the classiﬁcation of these topologies is well known. Any compact, connected, oriented two-dimensional surface without boundary is topologically equivalent to a sphere with g handles; g is known as the genus of the surface. In ﬁgure 3.10 we show g = 0, 1, and 2. Boundaries can be added by cutting holes in a closed surface. Any compact, connected, oriented two-dimensional surface is topologically equivalent to a sphere with g handles and b holes. For example, (g, b) = (0, 1) is a disk, (0, 2) is an annulus, and (0, 3) is a pair of pants. To describe unoriented surfaces, it is useful to introduce the crosscap: cut a hole in the surface and identify diametrically opposite points. In more detail, take a complex coordinate z, cut out a disk of slightly less than unit radius, and glue the opposite points together by deﬁning pairs of points with z = −1/¯z to be equivalent. The resulting surface is unoriented because the gluing is antiholomorphic. Also, unlike the case of a boundary, the cross-cap introduces no edge or other local feature: the edge introduced by the cutting is only the boundary of a coordinate patch. Any compact connected closed surface, oriented or unoriented, can be obtained by adding g handles and c cross-caps to the sphere; any compact connected surface can be obtained by adding g handles, b boundaries, and c cross-caps to the sphere. Actually, these descriptions

3.6 Vertex operators

101

are somewhat redundant: in either the case with boundary or the case without, one obtains every topology exactly once by restricting the sum to let only one of g and c be nonzero. For example, (g, b, c) = (0, 0, 1) is the projective plane, (g, b, c) = (0, 1, 1) is the M¨ obius strip, (g, b, c) = (0, 0, 2) is the Klein bottle, and a torus with cross-cap can be obtained either as (g, b, c) = (0, 0, 3) or as (g, b, c) = (1, 0, 1), trading two cross-caps for a handle. The Euler number is given by χ = 2 − 2g − b − c .

(3.5.6)

The number of distinct topologies is very small compared to the number of distinct Feynman graphs at a given order in ﬁeld theory. For example, in the closed oriented theory, there is exactly one topology at each order of perturbation theory. In a ﬁeld theory, the number of graphs grows factorially with the order. The single string graph contains all the ﬁeld theory graphs. In various limits, where handles become long compared to their circumferences, we can approximate the handles by lines and the string graph in that limit is approximated by the corresponding Feynman graph.

3.6

Vertex operators

Using the state–operator mapping, the vertex operator for the closed string tachyon is

V0 = 2gc → gc

d2 σ g 1/2 eik·X d2 z : eik·X : .

(3.6.1)

We have now included a factor gc in the mapping. This is the closed string coupling constant, from adding an extra string to the process: we use the normalization of the vertex operator as the deﬁnition of the coupling. In the second line we have gone to the ﬂat world-sheet. The vertex operator must be (diﬀ×Weyl)-invariant, so in particular it must be conformally invariant on the ﬂat world-sheet. To oﬀset the transformation of d2 z, the operator must be a tensor of weight (1, 1). By a straightforward OPE calculation, eik·X is a tensor of weight h = ˜h = α k 2 /4, so the condition is m2 = −k 2 = −

4 . α

(3.6.2)

This is precisely the mass found in the light-cone quantization. Similarly, the tensor states at the ﬁrst excited level of the closed string

102

3 The Polyakov path integral

have the ﬂat world-sheet vertex operators

2gc ¯ ν eik·X : . d2 z : ∂X µ ∂X (3.6.3) α The normalization relative to the tachyon vertex operator follows from the state–operator mapping, and we will see in chapter 6 that the same relative normalization is obtained from unitarity of the S-matrix. The weight is α k 2 h = ˜h = 1 + , (3.6.4) 4 so these are massless, again agreeing with the light-cone quantization. There are further conditions in order that this be a tensor, as we will work out below. Vertex operators in the Polyakov approach We have in the state–operator mapping a systematic means of writing the ﬂat world-sheet vertex operator for any state. Since the world-sheet can always be made locally ﬂat, this is in principle all we need. For perspective, however, it is useful to study the curved world-sheet vertex operators in the Polyakov formalism. We will do this in the remainder of the section. The method we use will be somewhat clumsy and not as systematic as the state–operator mapping, but some of the results are useful. The reader need not work out all the details. Any operator to be included in the Polyakov path integral (3.5.5) must respect the local diﬀ×Weyl symmetry of the theory. We have not been speciﬁc in the ﬁrst line of eq. (3.6.1) as to how the operator is to be deﬁned on the curved world-sheet. As in the earlier discussion of the Weyl anomaly, it is convenient to use a method that preserves diﬀ invariance automatically, and to check the Weyl transformation by hand. For most purposes dimensional regularization is used in the literature. However, we will not have extensive need for this and so will not introduce it; rather, we will generalize the normal ordering introduced earlier. Deﬁne a renormalized operator [ F ]r by

1 [ F ]r = exp 2

δ δ d σd σ ∆(σ, σ ) µ F. δX (σ) δXµ (σ ) 2

2

(3.6.5)

Here α (3.6.6) ln d2 (σ, σ ) , 2 where d(σ, σ ) is the geodesic distance between points σ and σ . As with normal ordering, the expression (3.6.5) instructs us to sum over all ways of ∆(σ, σ ) =

103

3.6 Vertex operators

contracting pairs in F using ∆(σ, σ ). On the ﬂat world-sheet, d2 (σ, σ ) = |z − z |2 and this reduces to conformal normal ordering, which we have studied in detail. On a curved world-sheet it cancels the singular part of the self-contractions of ﬁelds in F. Diﬀ invariance is obvious, but the contraction depends on the metric and so introduces the Weyl dependence 1 2

δ δ [ F ]r , µ δX (σ) δXµ (σ ) (3.6.7) the ﬁrst term being the explicit Weyl dependence in the operator. A vertex operator for a state of momentum k µ must transform under translations in the same way as the state, and so will be of the form eik·X times derivatives of X µ . Operators with diﬀerent numbers of derivatives will not mix under Weyl transformations, so we can start by considering zero derivatives, the operator (3.6.1). The Weyl variation comes from the explicit factor of g 1/2 as well as from the renormalization as given in eq. (3.6.7), δW [ F ] r = [ δW F ] r +

δW V0 = 2gc

d2 σd2 σ δW ∆(σ, σ )

d2 σ g 1/2 2δω(σ) −

k2 δW ∆(σ, σ) [ eik·X(σ) ]r . 2

(3.6.8)

At short distance, d2 (σ, σ ) ≈ (σ − σ )2 exp(2ω(σ))

(3.6.9)

and so α ln(σ − σ )2 . 2 The Weyl variation is nonsingular in the limit σ → σ, ∆(σ, σ ) ≈ α ω(σ) +

(3.6.10)

δW ∆(σ, σ) = α δω(σ) . The condition for the Weyl variation (3.6.8) to vanish is then as already deduced above.

(3.6.11) k2

= 4/α ,

Oﬀ-shell amplitudes? Observe that a naive attempt to deﬁne oﬀ-shell amplitudes, by taking k µ oﬀ the mass shell, is inconsistent with the local world-sheet symmetries. In position space this means that a local probe of the string world-sheet, given by inserting δ (X(σ) − x0 ) = D

dD k exp[ik · (X(σ) − x0 )] (2π)D

(3.6.12)

into the path integral, is inconsistent because it involves all momenta. From several points of view this is not surprising. First, the argument given above that we could use pointlike sources (vertex operators) for

104

3 The Polyakov path integral

string states required a limiting process that essentially restricts us to on-shell questions. Second, string theory contains gravity, and in general relativity simple oﬀ-shell amplitudes do not exist. This is because we have to specify the location of the probe, but the coordinates are unphysical and coordinate-invariant oﬀ-shell quantities are much more complicated. Third, in string theory we do not have the freedom to introduce additional ﬁelds to measure local observables (analogous to the way electroweak processes are used to probe strongly interacting systems): we must use the strings themselves, or other objects that we will discuss (D-branes and solitons) that are intrinsic to the theory. One can deﬁne oﬀ-shell amplitudes, at least in perturbation theory, if one ﬁxes a gauge (light-cone gauge being simplest for this) and uses stringy sources. The S-matrix formula (3.5.5) can then be derived in a way more akin to the reduction formula from quantum ﬁeld theory, where it is possible to deﬁne ﬁnite-time transition amplitudes and then take the inﬁnite-time limit. Incidentally, the reader who is trying to relate the present discussion to ﬁeld theory should note that the path integral expression (3.5.5) is not analogous to a Green’s function but will in fact have the properties of an S-matrix element. The analogous object in ﬁeld theory is the Green’s function with its external propagators amputated and its external momenta restricted to the mass shell. The discussion of the local probe (3.6.12) also shows the problem with a contact interaction between strings. The simplest form for such an interaction would be

2

d σ g(σ) M

1/2

M

d2 σ g(σ )1/2 δ D (X(σ) − X(σ )) ,

(3.6.13)

which is nonvanishing whenever the world-sheet self-intersects. However, the delta function involves all momenta, as in eq. (3.6.12), so this is not Weyl-invariant. The structure of string theory is quite rigid: strings can couple to strings only in the way we have described at the beginning of this chapter. Massless closed string vertex operators It is interesting, if a little messy, to carry the Polyakov treatment of vertex operators one level further. There is no way to make a world-sheet scalar with exactly one derivative, so the next case is two derivatives. The diﬀ-invariant possibilities are gc V1 = α

#

d2 σ g 1/2 (g ab sµν + iab aµν )[ ∂a X µ ∂b X ν eik·X ]r $

+ α φR [ eik·X ]r ,

(3.6.14)

3.6 Vertex operators

105

where sµν , aµν , and φ are respectively a symmetric matrix, an antisymmetric matrix, and a constant. The antisymmetric tensor ab is normalized g 1/2 12 = 1, or g 1/2 z¯z = −i. The i accompanying the antisymmetric tensor in the vertex operator can be understood as arising from the Euclidean continuation, because this term necessarily has an odd number of time derivatives (one) . This result applies generally to Euclidean actions and vertex operators. To extend the analysis of Weyl invariance we need to work out the Weyl dependence (3.6.11) of the geodesic distance to higher order. One ﬁnds 1 (3.6.15a) ∂a δW ∆(σ, σ ) = α ∂a δω(σ) , σ =σ 2 1+γ ∂a ∂b δW ∆(σ, σ ) = (3.6.15b) α ∇a ∂b δω(σ) , σ =σ 2 γ ∇a ∂b δW ∆(σ, σ ) = − α ∇a ∂b δω(σ) . (3.6.15c) σ =σ 2 Here, γ = − 23 , but we have left it as a parameter for later reference; the third equation can be obtained from the second and the gradient of the ﬁrst. Using eq. (3.3.5) for the variation of the curvature and eqs. (3.6.7) and (3.6.15) for the variation of the renormalization gives # gc d2 σ g 1/2 δω (g ab Sµν + iab Aµν )[ ∂a X µ ∂b X ν eik·X ]r δW V1 = 2 $ +α FR [ eik·X ]r ,

(3.6.16)

where Sµν = −k 2 sµν + kν k ω sµω + kµ k ω sνω − (1 + γ)kµ kν sωω + 4kµ kν φ , (3.6.17a) 2 ω ω Aµν = −k aµν + kν k aµω − kµ k aνω , (3.6.17b) 1 1 F = (γ − 1)k 2 φ + γk µ k ν sµν − γ(1 + γ)k 2 sνν . (3.6.17c) 2 4 In deriving the Weyl variation (3.6.16) we have integrated by parts and also used the relation α γ [ ∇2 X µ eik·X ]r = i k µ R [ eik·X ]r . (3.6.18) 4 The left-hand side would vanish by the naive equation of motion, but here the equation of motion multiplies another operator at the same point. The general principle that holds in this case is that the operator need not vanish but it is not independent — it can be expanded in terms of the other local operators in the theory. In general the coeﬃcients of these operators depend on the renormalization scheme; the result (3.6.18) can be obtained by taking the Weyl variation of each side.

106

3 The Polyakov path integral

Since Sµν , Aµν , and F multiply independent operators, the condition for Weyl invariance is Sµν = Aµν = F = 0 .

(3.6.19)

In order to get an accurate count of the number of operators, we must note that not all V1 of the form (3.6.14) are independent. Rather, under (3.6.20a) sµν → sµν + ξµ kν + kµ ξν , aµν → aµν + ζµ kν − kµ ζν , (3.6.20b) γ (3.6.20c) φ → φ+ k·ξ , 2 the change in V1 integrates to zero using the equation of motion (3.6.18). Choose now a vector nµ satisfying n · k = 1 and n2 = 0. A complete set of independent operators is obtained by restricting to nµ sµν = nµ aµν = 0 .

(3.6.21)

This is 2D equations for the 2D parameters ξµ and ζµ ; one can show that one equation and one parameter are trivial, but that eqs. (3.6.21) are otherwise nondegenerate and so deﬁne an independent set of vertex operators. By using condition (3.6.21) and solving ﬁrst Sµν nµ nν = 0, then Sµν nµ = Aµν nµ = 0, and ﬁnally the full Sµν = Aµν = F = 0, one ﬁnds that k2 = 0 , (3.6.22a) ν µ k sµν = k aµν = 0 , (3.6.22b) 1+γ µ φ= (3.6.22c) sµ . 4 Again we ﬁnd a mass-shell condition (3.6.22a), this time corresponding to the ﬁrst excited states of the closed string. There is also the condition (3.6.22b) that the polarization be transverse to the momentum, as required for a physical polarization of a massless tensor ﬁeld. In the frame in which 1 k µ = (1, 1, 0, 0, . . . , 0) , nµ = (−1, 1, 0, 0, . . . , 0) , (3.6.23) 2 conditions (3.6.21) and (3.6.22b) imply that all 0- and 1-components vanish, while eq. (3.6.22c) ﬁxes φ. This leaves precisely (D − 2)2 independent operators, the same number of massless closed string states as found in the light-cone gauge. The condition nµ sµν = nµ aµν = 0 is needed to remove zero-norm lightlike polarizations. However, the introduction of the vector nµ spoils manifest Lorentz invariance, and the theory is Lorentz-invariant only because diﬀerent choices of nµ are equivalent due to (3.6.20). The equivalence relations (3.6.20) are therefore essential in order for the theory to be

107

3.6 Vertex operators

Lorentz-invariant with a positive Hilbert space norm. These relations are the signatures of local spacetime symmetries. In particular, ξµ is an inﬁnitesimal spacetime coordinate transformation, and ζµ is the local spacetime symmetry of the antisymmetric tensor ﬁeld. We now see in the interacting theory the existence of local spacetime symmetries, which we argued in section 1.4 had to be present because of the existence of massless spin-1 and spin-2 ﬁelds. There is a technical point we should mention. Diﬀerent renormalization schemes assign diﬀerent names to the renormalized operators. The most common scheme is dimensional regularization; the operators in this scheme are related to those in the scheme above by [ eik·X ]DR = [ eik·X ]r , [ ∂a X µ eik·X ]DR = [ ∂a X µ eik·X ]r ,

(3.6.24a) (3.6.24b)

[ ∂a X µ ∂b X ν eik·X ]DR = [ ∂a X µ ∂b X ν eik·X ]r − [ ∇a ∂b X µ eik·X ]DR = [ ∇a ∂b X µ eik·X ]r + i

α gab η µν R[ eik·X ]r , 12 (3.6.24c)

α gab k µ R[ eik·X ]r . 12

(3.6.24d)

Comparing with eq. (3.6.18), we see that in dimensional regularization γ = 0, so the equations of motion hold inside the regulated operator. This is a convenience — many equations simplify. In particular, in the spacetime gauge transformation (3.6.20), φ is invariant. Open string vertex operators The extension to the open string is straightforward. We leave the details for the reader as exercise 3.9, giving only some results. The tachyon vertex operator is

go

∂M

ds [ eik·X ]r

(3.6.25)

and is Weyl-invariant for k 2 = 1/α . The photon vertex operator is go − i 1/2 eµ (2α )

∂M

˙ µ eik·X ]r , ds [ X

(3.6.26)

where the normalization relative to the tachyon is obtained either from the state–operator mapping or unitarity, the i is as discussed for the closed string vertex operator (3.6.14), and the sign is a matter of convention. This operator is Weyl-invariant if k 2 = 0 and k · e = 0. There is an equivalence eµ ∼ = eµ + λkµ , which is a spacetime gauge transformation. This leaves D − 2 transverse polarizations.

108

3 The Polyakov path integral 3.7

Strings in curved spacetime

Now we are ready to consider an important new issue, strings moving in a curved spacetime. Let us recall ﬁrst that the point-particle action has a natural extension to curved spacetime. Replacing the ﬂat metric ηµν with a general metric Gµν (x) gives the action Spp =

1 2

˙ ν − ηm2 . ˙ µX dτ η −1 Gµν (X)X

(3.7.1)

After eliminating η this becomes the invariant proper time along the particle world-line. Its variation is well known to give the geodesic equation, representing the motion of a particle in a gravitational ﬁeld. Making the same replacement in the Polyakov action gives Sσ =

1 4πα

M

d2 σ g 1/2 g ab Gµν (X)∂a X µ ∂b X ν .

(3.7.2)

This is a natural guess, but the reader might make the following objection. We have learned that the graviton itself is one state of the string. A curved spacetime is, roughly speaking, a coherent background of gravitons, and therefore in string theory it is a coherent state of strings. Merely putting a curved metric in the action (3.7.2) doesn’t seem stringy enough. To see why it is justiﬁed, consider a spacetime that is close to ﬂat, Gµν (X) = ηµν + χµν (X)

(3.7.3)

with χµν small. The integrand in the world-sheet path integral is

1 exp(−Sσ ) = exp(−SP ) 1 − 4πα

2

d σg M

1/2 ab

µ

ν

g χµν (X)∂a X ∂b X + . . .

.

(3.7.4) The term of order χ is precisely the vertex operator for the graviton state of the string, eq. (3.6.14) with χµν (X) = −4πgc eik·X sµν .

(3.7.5)

So indeed, inserting the curved metric does seem to make contact with what we already know. The action (3.7.2) can be thought of as describing a coherent state of gravitons by exponentiating the graviton vertex operator. This suggests a natural generalization: include backgrounds of the other massless string states as well. From the form of the vertex operators, eq. (3.6.14), one has Sσ =

1 4πα

M

d2 σ g 1/2

g ab Gµν (X) + iab Bµν (X) ∂a X µ ∂b X ν + α RΦ(X) . (3.7.6)

3.7 Strings in curved spacetime

109

The ﬁeld Bµν (X) is the antisymmetric tensor, and the dilaton involves both Φ and the diagonal part of Gµν , as implied by the equation of motion (3.6.22c). As a check, let us see that the spacetime gauge invariances are respected. Under a change of variables in the path integral corresponding to a ﬁeld redeﬁnition X µ (X), the action (3.7.6) is invariant with Gµν and Bµν transforming as tensors and Φ as a scalar. From the spacetime point of view, this is a coordinate transformation. The action is also invariant under δBµν (X) = ∂µ ζν (X) − ∂ν ζµ (X) ,

(3.7.7)

which adds a total derivative to the Lagrangian density. This is a generalization of the electromagnetic gauge transformation to a potential with two antisymmetric indices. The three-index ﬁeld strength Hωµν = ∂ω Bµν + ∂µ Bνω + ∂ν Bωµ

(3.7.8)

is invariant. There is a similar generalization to an antisymmetric n-tensor potential, which plays an important role in the superstring and will be developed in volume two. The theory thus depends only on gauge-invariant objects built out of the metric and other ﬁelds. One can think of the ﬁelds X µ as being coordinates on a manifold. This is called the target space, because the X µ deﬁne an embedding, world-sheet → target .

(3.7.9)

In string theory, the target space is spacetime itself. A ﬁeld theory such as (3.7.2), in which the kinetic term is ﬁeld-dependent and so ﬁeld space is eﬀectively a curved manifold, is known for historic reasons as a nonlinear sigma model. Nonlinear sigma models have had many applications in particle physics and quantum ﬁeld theory. For example, the neutral and charged pion ﬁelds can be regarded to good approximation as coordinates on the group manifold S U(2). The nonlinear sigma model action is no longer quadratic in X µ , so the path integral is now an interacting two-dimensional quantum ﬁeld theory. Expand the path integral around the classical solution X µ (σ) = xµ0 for any chosen point xµ0 . With X µ (σ) = xµ0 + Y µ (σ),

Gµν (X)∂a X µ ∂b X ν = Gµν (x0 ) + Gµν,ω (x0 )Y ω

+ 12 Gµν,ωρ (x0 )Y ω Y ρ + . . . ∂a Y µ ∂b Y ν .

(3.7.10)

The ﬁrst term in this expansion is a quadratic kinetic term for the ﬁeld Y µ . The next term is a cubic interaction, and so forth. The coupling constants in the expansion, Gµν,ω (x0 ) and so on, involve derivatives of

110

3 The Polyakov path integral

the metric at x0 . In a target space with characteristic radius of curvature Rc , derivatives of the metric are of order Rc−1 . The eﬀective dimensionless coupling in the theory is therefore α1/2 Rc−1 . If the radius of curvature Rc is much greater than the characteristic length scale of the string, then this coupling is small and perturbation theory in the two-dimensional ﬁeld theory is a useful tool. Notice that in this same regime we can employ another tool as well. Since the characteristic length scale is long compared to the string, we can ignore the internal structure of the string and use low energy eﬀective ﬁeld theory. String theory enters in determining the eﬀective low energy action for the ﬁelds, as we will soon see. We have also implicitly used α1/2 Rc−1 1 when restricting our attention to massless backgrounds: when wavelengths are long compared to the string scale, massive string states are not created. The nonlinear sigma model (3.7.2) is a renormalizable theory: the ﬁelds Y µ have dimension zero and so the interactions all have dimension two. Nevertheless the couplings, which are the coeﬃcients in the expansion of the metric, are inﬁnite in number (unless some symmetry restricts the form of the metric). In fact it is more useful to think of the metric function itself, rather than its expansion coeﬃcients, as deﬁning the couplings of the theory.

Weyl invariance We have emphasized that Weyl invariance is essential to the consistency of string theory. The action (3.7.6) will deﬁne a consistent string theory only if the two-dimensional quantum ﬁeld theory is Weyl-invariant. This action happens to be the most general action that is classically invariant under a rigid Weyl transformation, with δω independent of σ. This is easy to see: under a rigid Weyl transformation, the scaling of a term with n derivatives will be proportional to 2 − n. Of course, it is necessary also to consider local Weyl transformations, under which the Gµν and Bµν terms in the action (3.7.6) are classically invariant while the Φ term is not, and to include the quantum contributions to the Weyl transformation properties. In the limit that Bµν and Φ are small, and Gµν is close to ηµν , we can use the results of the previous section to ﬁnd the Weyl transformation. Write Sσ = SP − V1 + . . ., with SP the ﬂat-space action and V1 the vertex operator (3.6.14). Then Gµν (X) = ηµν − 4πgc sµν eik·X ,

(3.7.11a)

Bµν (X) = −4πgc aµν eik·X ,

(3.7.11b)

Φ(X) = −4πgc φe

ik·X

.

(3.7.11c)

3.7 Strings in curved spacetime

111

We could of course take linear combinations of vertex operators with diﬀerent momenta. To ﬁrst order the Weyl variation is then given in eq. (3.6.16); we take for convenience a renormalization scheme in which γ = 0. Relating this Weyl transformation to T aa as in eq. (3.4.6) gives 1 G ab i B ab 1 βµν g ∂a X µ ∂b X ν − βµν ∂a X µ ∂b X ν − β Φ R , 2α 2α 2 where, to linear order in χµν , Bµν and Φ, T aa = −

(3.7.12)

α 2 ∂ χµν − ∂ν ∂ω χµω − ∂µ ∂ω χων + ∂µ ∂ν χωω + 2α ∂µ ∂ν Φ , 2 (3.7.13a) α ω B βµν ≈ − ∂ Hωµν , (3.7.13b) 2 D − 26 α 2 βΦ ≈ (3.7.13c) − ∂ Φ. 6 2 We have included in β Φ the ﬂat spacetime anomaly found in section 3.4, including the contribution proportional to 26 from the ghost ﬁelds. The symbol β is used for the coeﬃcients in T aa because these are essentially the renormalization group beta functions, governing the dependence of the physics on world-sheet scale. We will discuss this connection further, though it does not come up in detail until chapter 15. The Weyl anomalies (3.7.13) have further contributions from higher orders in the ﬁelds. For example, expanding the path integral to second order in the cubic interaction from the expansion (3.7.10) will give rise to divergences when the two interaction vertices approach one another. ¯ ν times the Their OPE includes a singularity of the form |z|−2 ∂X µ ∂X 2 square of the derivative of Gαβ . Integrating d z produces a logarithmic divergence. Diﬀ invariance requires that the integral be cut oﬀ in terms of the invariant distance, |z| exp(ω) > a0 . This introduces a dependence G on the scale of the metric, ln zmin = −ω + ln a0 . The Weyl variation βµν 2 thus gets a contribution proportional to O(Gαβ,γ ) which combines with the linear second derivative term to form the spacetime Ricci tensor. We will not work this out in detail, but quote the result keeping all terms with up to two spacetime derivatives G βµν ≈−

G = α Rµν + 2α ∇µ ∇ν Φ − βµν

α Hµλω Hν λω + O(α2 ) , 4

(3.7.14a)

α B βµν = − ∇ω Hωµν + α ∇ω ΦHωµν + O(α2 ) , (3.7.14b) 2 D − 26 α 2 α βΦ = − ∇ Φ + α ∇ω Φ∇ω Φ − Hµνλ H µνλ + O(α2 ) . 6 2 24 (3.7.14c) Several terms in (3.7.14) can be recognized from the linear approximation

112

3 The Polyakov path integral

(3.7.13), but now made covariant under a change of spacetime coordinates. The spacetime Ricci tensor is Rµν , distinguished from the world-sheet Ricci tensor Rab . Terms with more derivatives are higher order in the worldsheet α1/2 Rc−1 expansion. Incidentally, the most eﬃcient method to carry the calculation of the Weyl anomaly to higher order is not the one we have described, but rather dimensional regularization. The condition that the world-sheet theory be Weyl-invariant is thus G B = βµν = βΦ = 0 . βµν

(3.7.15)

G = 0 These are sensible-looking equations of motion. The equation βµν resembles Einstein’s equation with source terms from the antisymmetric B = 0 is the antisymmetric tensor ﬁeld and the dilaton. The equation βµν tensor generalization of Maxwell’s equation, determining the divergence of the ﬁeld strength.

Backgrounds One other qualitative feature of the ﬁeld equations is readily understood: the ﬁeld Φ always occurs diﬀerentiated, so there is an invariance under X µ -independent shifts of Φ. This is because such a shift only changes the world-sheet action (3.7.6) by a term proportional to the Euler number, and so does not aﬀect local properties like Weyl invariance. In particular, the background Gµν (X) = ηµν ,

Bµν (X) = 0 ,

Φ(X) = Φ0

(3.7.16)

is exactly Weyl-invariant for any constant Φ0 . This is just the ﬂat spacetime action (3.2.2), with λ = Φ0 .

(3.7.17)

The value of λ determines the coupling strength between strings, but we now see that this does not mean that there are diﬀerent string theories with diﬀerent values of this parameter. Diﬀerent values of λ correspond not to diﬀerent theories, but to diﬀerent backgrounds in a single theory. The other parameter appearing in the Nambu–Goto and Polyakov actions, the Regge slope α , is also not really a free parameter because it is dimensionful. It simply deﬁnes the unit of length, and can be absorbed in the deﬁnition of X µ. Changing the ﬁelds Gµν , Bµν , and Φ in the world-sheet action would seem, from the two-dimensional point of view, to give a new theory. From the point of view of the full string theory, one is merely looking at a diﬀerent background — a diﬀerent state — in the same theory. This is one of the features that makes string theory attractive. In the Standard Model, and in most attempts to unify the Standard Model within

3.7 Strings in curved spacetime

113

quantum ﬁeld theory, there are many constants that are not determined by the theory; any values for these constants give a consistent theory. In string theory there are no such free parameters — the coupling constants depend on the state and are determined in principle by the dynamics. Of course, for now this only moves the diﬃculty elsewhere, because we do not understand the dynamics well enough to know the values of the background ﬁelds. It is striking that Einstein’s equation turns up in what seems to be a rather out-of-the-way place, as the condition for Weyl invariance of a two-dimensional ﬁeld theory. String theory provides the physical connection between these two ideas. It is also notable that, because of the need for Weyl invariance, strings can propagate consistently only in a background that satisﬁes appropriate ﬁeld equations. This parallels our earlier discovery that only on-shell vertex operators make sense. Another observation: the condition D = 26 came from the R term in T aa . In the nonlinear sigma model this is generalized to β Φ = 0. We see in eq. (3.7.14c) that the leading term in β Φ is proportional to D − 26 but that there are corrections involving the gradients of the ﬁelds. Evidently we can have other values of D if the ﬁelds are not constant. This is true, although we cannot really conclude it from eq. (3.7.14c) because this has been derived in the approximation α1/2 Rc−1 1 where the correction terms in β Φ are small. In fact, exact solutions with D = 26 are known. For now we give one simple example, saving others for later. This is Gµν (X) = ηµν ,

Bµν (X) = 0 ,

Φ(X) = Vµ X µ .

(3.7.18)

The beta functions (3.7.14) vanish if Vµ V µ =

26 − D . 6α

(3.7.19)

This result is actually exact, because the ﬁelds (3.7.18) are constant or linear in X µ , so the world-sheet path integral remains Gaussian. Varying gab to determine Tab for this theory, one ﬁnds that it is none other than the linear dilaton CFT (hence the name), whose central charge c = D + 6α Vµ V µ is indeed 26 under the condition (3.7.19). Because Φ must have a large gradient in order to change D, this CFT does not describe our roughly static and homogeneous four-dimensional spacetime, but it may have cosmological applications. This CFT is also important because the cases D = 1 and D = 2 are simple enough for the string theory to be exactly solvable, as we will discuss in section 9.9.

114

3 The Polyakov path integral The spacetime action

The ﬁeld equations (3.7.15) can be derived from the spacetime action 1 S= 2 2κ0

D

d x (−G)

1/2 −2Φ

e

−

2(D − 26) 1 + R − Hµνλ H µνλ 3α 12

+ 4∂µ Φ∂µ Φ + O(α ) .

(3.7.20)

The normalization constant κ0 is not determined by the ﬁeld equations and has no physical signiﬁcance since it can be changed by a redeﬁnition of Φ. One can verify that δS = −

1 2κ20 α

dD x (−G)1/2 e−2Φ δGµν β Gµν + δBµν β Bµν

+ 2δΦ − 12 Gµν δGµν (β Gωω − 4β Φ ) .

(3.7.21)

The action is written in bold as a reminder that this is the eﬀective action governing the low energy spacetime ﬁelds, unlike the world-sheet actions that have appeared everywhere previously in this book. It is often useful to make a ﬁeld redeﬁnition of the form ˜ µν (x) = exp(2ω(x))Gµν (x) , G

(3.7.22)

which is a spacetime Weyl transformation. The Ricci scalar constructed ˜ µν is from G

˜ = exp(−2ω) R − 2(D − 1)∇2 ω − (D − 2)(D − 1)∂µ ω∂µ ω . R

(3.7.23)

For the special case D = 2, this is the Weyl transformation (3.3.5). Let ω = 2(Φ0 − Φ)/(D − 2) and deﬁne ˜ = Φ − Φ0 , Φ

(3.7.24)

which has vanishing expectation value. The action becomes

˜ ˜ ˜ 1/2 − 2(D − 26) e4Φ/(D−2) d X (−G) +R 3α 1 −8Φ/(D−2) 4 ˜ µνλ µ˜ ˜ ˜ ˜ − e Hµνλ H − ∂µ Φ∂ Φ + O(α ) , 12 D−2

1 S= 2 2κ

D

(3.7.25)

where tildes have been inserted as a reminder that indices here are raised ˜ µν , the gravitational Lagrangian density takes ˜ µν . In terms of G with G 2 . The constant κ = κ eΦ0 is the ˜ ˜ 1/2 R/2κ the standard Hilbert form (−G) 0 gravitational coupling, which in four-dimensional gravity has the value κ = (8πGN )1/2 =

(8π)1/2 = (2.43 × 1018 GeV)−1 . MP

(3.7.26)

3.7 Strings in curved spacetime

115

˜ µν Commonly, Gµν is called the sigma model metric or string metric, and G the Einstein metric. Because of the force from dilaton exchange, there is no equivalence principle and so no way to single out a preferred deﬁnition of the metric. Looking ahead, there are higher order eﬀects in string theory that can give the dilaton a mass and so a ﬁnite range — a good thing, because the equivalence principle does of course hold to good accuracy. Note that the dilaton appears in the action (3.7.20) in an overall factor e−2Φ and otherwise is always diﬀerentiated. This is consistent with the fact that adding an overall constant to the dilaton has no eﬀect on the Weyl anomaly. From the spacetime point of view this multiplies the action by a constant, which does not change the equations of motion. It also reﬂects the fact that in quantum ﬁeld theory it is possible to rescale the ﬁelds in such a way that the coupling g appears only as an overall factor of g −2 , which in string theory is e−2Φ . For example, the Yang–Mills action is usually introduced in the form

1 dD x Tr (Fµν F µν ) , S =− 4 Fµν = ∂µ Aν − ∂ν Aµ − ig[Aµ , Aν ]

(3.7.27a) (3.7.27b)

(matrix notation). By deﬁning gAµ = Aµ ,

gFµν = Fµν ,

(3.7.28)

g is removed from the ﬁeld strength, covariant derivative, and gauge transformation and appears only in an overall g −2 in the action. In string theory this holds when the action is written in terms of the ﬁelds appearing on the string world-sheet. If we make a dilaton-dependent ﬁeld redeﬁnition as in going to the form (3.7.25) then it no longer holds. When g appears only in the normalization of the action, every propagator contains a factor of g 2 and every interaction a factor of g −2 . It is not hard to show that an L-loop contribution to the eﬀective action has exactly L − 1 more propagators than vertices and so scales as g 2(L−1) . In string theory this is e−χΦ , as we would expect. The discussion of backgrounds extends to open strings. We have seen that open string vertex operators are integrated along the boundary, so open string ﬁelds appear as boundary terms in the sigma model action. Compactiﬁcation and CFT The four string theories studied thus far (with or without boundaries, oriented or unoriented) have a number of features in common. One good feature is the automatic inclusion of general relativity. There are also several bad features: the need for 26 dimensions, the existence of the tachyon, and the absence of fermions in the spectrum. The elimination

116

3 The Polyakov path integral

of the tachyon and the introduction of fermions will have to wait until volume two. However, the presence of general relativity provides a natural way to account for the extra dimensions. In general relativity the geometry of spacetime is dynamical, not ﬁxed. Flat spacetime is only one of many solutions to the ﬁeld equations. There may well be solutions in which some dimensions are large and ﬂat and others are small and highly curved. The metric would be

gMN =

ηµν 0

0 gmn (xp )

.

(3.7.29)

We have divided the 26 coordinates M, N = 0, . . . , 25 into four ‘spacetime’ coordinates µ, ν = 0, . . . , 3 and 22 ‘internal’ coordinates m, n, p = 4, . . . , 25. The metric (3.7.29) is ﬂat in the four spacetime dimensions and curved in the 22 internal dimensions. The internal dimensions are assumed to be compact. As one example, the background ﬁeld equations (3.7.15) are satisﬁed to order α if the dilaton ﬁeld is constant, the antisymmetric tensor is zero, and the metric for the internal space is Ricci-ﬂat, Rmn = 0. In other words, spacetime is of the form M d × K, where M d is ddimensional Minkowski space and K is some (26−d)-dimensional compact Riemannian space.4 In such a spacetime, the physics on length scales much longer than the size of K is the same as in a d-dimensional Minkowski space. One says that 26 − d dimensions have been compactiﬁed, though cosmologically it seems more appropriate to say that the large dimensions have decompactiﬁed. As we discussed brieﬂy in the introduction, there are good reasons, independent of string theory, to consider the possibility that our spacetime might be of this form. The compact dimensions must be small enough not to have been observed. In fact, we will see in chapter 8, and in more detail in chapter 18, that the size of any extra dimensions must be within a few orders of magnitude of the Planck length, 1.6 × 10−33 cm. We can also think about generalizing this idea further. The basic physical consistency conditions are few in number. We require Lorentz invariance, and we require that quantum mechanical probabilities be positive and conserved — to date, nothing in string theory seems to require that the standard quantum mechanical framework be modiﬁed, though it would be interesting to learn otherwise. Although these conditions are simple there is a tension between them, in gauge theory and gravitation as well as in string theory. In a manifestly covariant gauge there are negative norm timelike excitations that must decouple from physical amplitudes. In the light-cone gauge the inner product is positive but covariance is no longer manifest. 4

We will generally use D to refer to the total dimension of a string theory and d when we may be talking about a subset of the world-sheet ﬁelds.

3.7 Strings in curved spacetime

117

One necessary condition is therefore two-dimensional diﬀ invariance, so that the unphysical oscillations are merely oscillations of the coordinate system. Weyl invariance is a more technical requirement: the extra degree of freedom that appears when Weyl invariance is lost is less intuitive than an oscillation of the coordinate system, but we have argued in section 3.4 that it is incompatible with Lorentz invariance. We will also make an additional technical assumption: that the embedding time X 0 appears in the world-sheet action only in the term −

1 4πα

M

d2 σ g 1/2 g ab ∂a X 0 ∂b X 0 .

(3.7.30)

For example, in the sigma model this means that the ﬁelds are static with G0µ = η0µ and B0µ = 0. This assumption is not much of a restriction. The form (3.7.30) is suﬃcient if one is interested in static states. Timedependent backgrounds are certainly of interest for cosmological and other reasons, but they can be analyzed using low energy eﬀective ﬁeld theory except in the extreme case that the time scale is of the order of the string scale. The reason for making the assumption (3.7.30) is to put the main problem, the wrong-sign nature of X 0 , in a rather explicit form. The requirement that the local invariances be maintained leads to the following proposal for a more general string theory: Replace the 25 spatial X µ ﬁelds with any unitary CFT having c = ˜c = 25. This ensures that the full two-dimensional theory, with X 0 and the ghosts, can be coupled to a curved metric in a (diﬀ×Weyl)-invariant way. The unitary (positive inner product) condition is necessary because there is only enough local symmetry to remove the unphysical X 0 excitation; the world-sheet inner product is relevant here because it becomes the spacetime inner product in the one-particle sector. We will see in volume two that further generalization is possible, enlarging simultaneously the local symmetry group and the set of unphysical excitations. If we are interested in a bosonic string theory whose low energy physics looks four-dimensional, we keep the free ﬁelds X µ for µ = 0, 1, 2, 3, and replace the remaining X µ with a compact unitary CFT having c = ˜c = 22. By compact, we mean that the spectrum is discrete, just as for a quantum ﬁeld theory in a ﬁnite volume. There have been some eﬀorts to construct all CFTs, or all CFTs having various additional properties. We will describe some of these in chapter 15. Incidentally, all of these theories have tachyons, the product of the unit state |1 from the internal CFT with the ground state |0; k from the four-dimensional CFT. How large a generalization have we just made? If the local symmetries are the only essential restriction, we are free to introduce ﬁelds on the world-sheet with various world-sheet and spacetime quantum numbers. This seems to take us far beyond the original theory of 26 X µ ﬁelds. How-

118

3 The Polyakov path integral

ever, in two dimensions there are many equivalences between seemingly diﬀerent quantum ﬁeld theories, so that the situation is not clear. It is generally assumed that all string theories based on diﬀerent CFTs but the same world-sheet gauge symmetries and topology are diﬀerent ground states of the same theory. This causes a semantic problem: we often refer to strings with diﬀerent world-sheet Lagrangians as diﬀerent theories, though they are most likely just diﬀerent vacua in one theory. We will take this even further in chapter 14, seeing that all string theories appear to be vacua of a single theory. As we develop the techniques of string theory in volume one, we will usually take the simplest case of ﬂat 26-dimensional spacetime in explicit examples. However, many results will be developed in such a way as to apply to the very general string theory we have described above. Exercises 3.1 (a) Evaluate the Euler number χ for a ﬂat disk and for a disk with the metric of a hemisphere. (b) Show that the Euler number of a sphere with b holes is 2 − b. You may take any convenient metric on the sphere. (c) Extend this to the result 2 − b − 2g for the sphere with b holes and g handles. 3.2 (a) Show that an n-index traceless symmetric tensor in two dimensions has exactly two independent components. The traceless condition means that the tensor vanishes when any pair of indices is contracted with g ab . (b) Find a diﬀerential operator Pn that takes n-index traceless symmetric tensors into (n + 1)-index traceless symmetric tensors. (c) Find a diﬀerential operator PnT that does the reverse. (d) For u an (n + 1)-index traceless symmetric tensor and v an n-index traceless symmetric tensor, show that with appropriate normalization (u, Pn v) = (PnT u, v). The inner product of traceless symmetric tensors of any rank is (t, t ) =

d2 σ g 1/2 t · t ,

the dot denoting contraction of all indices. 3.3 (a) Work out the form of the covariant derivative in conformal gauge. Show in particular that ∇z reduces to ∂z when acting on a tensor component with all ¯z indices, and vice versa. Since raising or lowering reverses z and ¯z indices, this means that it is always possible to work with simple derivatives by appropriately positioning the indices. (b) Use this to give simple expressions for Pn and PnT . 3.4 To gain familiarity with the Faddeev–Popov procedure, it is useful to

Exercises

119

consider it in a more general and abstract context. This is described in the ﬁrst page of section 4.2 (which is independent of section 4.1). Derive the general result (4.2.3). 3.5 (A continuation of the previous problem.) Consider two diﬀerent gauge choices, F A (φ) = 0 and GA (φ) = 0. The Faddeev–Popov procedure shows that the gauge-ﬁxed path integrals (4.2.3) are both equal to the original gauge-invariant path integral and so equal to each other. However, this passes through the somewhat heuristically deﬁned gauge-invariant path integral. Adapt the Faddeev–Popov procedure to show directly the equivalence of the two gauge-ﬁxed path integrals. This is what is relevant in practice, showing for example that manifestly covariant and manifestly unitary gauges deﬁne the same theory, which therefore has both properties. In chapter 4 we will show this equivalence by a more abstract method, BRST symmetry. 3.6 On dimensional grounds, a current that is conserved on a ﬂat worldsheet has the possible anomaly ∇a j a = aR . By comparing the Weyl transformations of the two sides, show that the ˜ ˜ is 4a. sum of the coeﬃcients of z −3 in T j and of ¯z −3 in T 3.7 In our second calculation of the Weyl anomaly, we have in a roundabout way calculated a Feynman graph with a loop of scalars or ghosts and with two external gravitational ﬁelds. Do the calculation for scalars directly, in momentum space. You will need to use your favorite invariant regulator: dimensional, Pauli–Villars, or zeta function. Show that the result agrees with eq. (3.4.19). The linearized expression for the curvature can be conveniently read from eqs. (3.7.13a) and (3.7.14a). (Warning: the full calculation is a bit lengthy.) 3.8 Verify the Weyl transformation (3.6.16) of the vertex operator V1 . 3.9 Find the Weyl invariance conditions for the open string tachyon and photon vertex operators. It is simplest to take a coordinate system in which the boundary is a straight line. Recall from exercise 2.10 that the propagator includes an image charge term, so it is eﬀectively doubled for ﬁelds directly on the boundary. 3.10 Find the general vertex operator and the conditions for Weyl invariance at the level m2 = 1/α of the open string. Find a complete set of independent Weyl-invariant operators and compare with the states found in the light-cone quantization. [Answer: there appears to be one extra operator. In fact its amplitudes vanish, but we need the formalism of the next chapter to understand the reason, that it corresponds to a null state.] G by the method suggested above 3.11 Calculate the order H 2 term in βµν

120

3 The Polyakov path integral

eq. (3.7.14). You need precisely the term (z¯z )−1 in the OPE; more singular terms require a shift in the tachyon background, but this is an artifact of the bosonic string. G = β B = 0, the nonlinear sigma model will be conformally 3.12 If βµν µν invariant on a ﬂat world-sheet. According to the discussion at the end of section 3.4, in this case β Φ must be independent of X. Show this for the form (3.7.14); you will need the Bianchi identities.

3.13 Consider a spacetime with d ﬂat dimensions and 3 dimensions in the shape of a 3-sphere. Let H be proportional to the completely antisymmetric tensor on the 3-sphere and let the dilaton be constant. Using the form (3.7.14) for the equations of motion, show that there are solutions with d + 3 = 26. These solutions are outside the range of validity of eq. (3.7.14), but we will see in chapter 15 that there are exact solutions of this form, though with H limited to certain quantized values. 3.14 Verify that the background (3.7.18) gives the linear dilaton energymomentum tensor (2.5.1).

4 The string spectrum

4.1

Old covariant quantization

In conformal gauge, the world-sheet ﬁelds are X µ and the Faddeev–Popov ghosts bab and ca . The Hilbert space is bigger than the actual physical spectrum of the string: the D sets of αµ oscillators include unphysical oscillations of the coordinate system, and there are ghost oscillators. As is generally the case in covariant gauges, there are negative norm states from the timelike oscillators (because the commutator is proportional to the spacetime metric ηµν ), and also from the ghosts. The actual physical space is smaller. To see how to identify this smaller space, consider the amplitude on the inﬁnite cylinder for some initial state |i to propagate to some ﬁnal state |f . Suppose that we have initially used the local symmetries to ﬁx the metric to a form gab (σ). Consider now a diﬀerent gauge, with metric gab (σ) + δgab (σ). A physical amplitude should not depend on this choice. Of course, for a change in the metric we know how the path integral changes. From the deﬁnition (3.4.4) of T ab ,

1 d2 σ g(σ)1/2 δgab (σ) f|T ab (σ)|i . (4.1.1) 4π In order that the variation vanish for arbitrary changes in the metric, we need δ f|i = −

ψ|T ab (σ)|ψ = 0

(4.1.2)

for arbitrary physical states |ψ , |ψ . There is another way to think about this. The original equation of motion from variation of gab was T ab = 0. After gauge-ﬁxing, this does not hold as an operator equation: we have a missing equation of motion because we do not vary gab in the gauge-ﬁxed theory. The condition (4.1.2) says that the missing equation of motion must hold for matrix elements between physical states. When we vary the gauge we must take into 121

122

4 The string spectrum

account the change in the Faddeev–Popov determinant, so the energymomentum tensor in the matrix element is the sum of the X µ and ghost contributions, g X + Tab . Tab = Tab

(4.1.3)

The X µ may be replaced by a more general CFT (which we will refer to as the matter CFT), in which case g m + Tab . Tab = Tab

(4.1.4)

In the remainder of this section we will impose the condition (4.1.2) in a simple but somewhat ad hoc way. This is known as the old covariant quantization (OCQ), and it is suﬃcient for many purposes. In the next section we will take a more systematic approach, BRST quantization. These are in fact equivalent, as will be shown in section 4.4. In the ad hoc approach, we will simply ignore the ghosts and try to restrict the matter Hilbert space so that the missing equation of motion m = 0 holds for matrix elements. In terms of the Laurent coeﬃcients Tab ˜m this is Lm n = 0, and in the closed string also Ln = 0. One might ﬁrst try m to require physical states to satisfy Ln |ψ = 0 for all n, but this is too strong; acting on this equation with Lm m and forming the commutator, one encounters an inconsistency due to the central charge in the Virasoro algebra. However, it is suﬃcient that only the Virasoro lowering and zero operator annihilate physical states, (Lm n + Aδn,0 )|ψ = 0

for n ≥ 0 .

(4.1.5)

For n < 0 we then have m

ψ|Lm n |ψ = L−n ψ|ψ = 0 .

(4.1.6)

m Lm† n = L−n

(4.1.7)

We have used

as follows from the Hermiticity of the energy-momentum tensor. The conditions (4.1.5) are consistent with the Virasoro algebra. At n = 0 we have as usual included the possibility of an ordering constant. A state satisfying (4.1.5) is called physical. In the terminology of eq. (2.9.8), a physical state is a highest weight state of weight −A. The condition (4.1.5) is similar to the Gupta–Bleuler quantization of electrodynamics. Using the adjoint (4.1.7), one sees that a state of the form |χ =

∞

Lm −n |χn

(4.1.8)

n=1

is orthogonal to all physical states for any |χn . Such a state is called spurious. A state that is both physical and spurious is called null. If |ψ

123

4.1 Old covariant quantization

is physical and |χ is null, then |ψ + |χ is also physical and its inner product with any physical state is the same as that of |ψ . Therefore these two states are physically indistinguishable, and we identify |ψ ∼ (4.1.9) = |ψ + |χ . The real physical Hilbert space is then the set of equivalence classes, HOCQ =

Hphys . Hnull

(4.1.10)

Let us see how this works for the ﬁrst two levels of the open string in ﬂat spacetime, not necessarily assuming D = 26. The only relevant terms are 2 Lm 0 = α p + α−1 · α1 + . . . ,

Lm ±1

1/2

= (2α )

p · α±1 + . . . .

(4.1.11a) (4.1.11b)

At the lowest mass level, the only state is |0; k . The only nontrivial 2 condition at this level is (Lm 0 + A)|ψ = 0, giving m = A/α . There are no null states at this level — the Virasoro generators in the spurious state (4.1.8) all contain raising operators. Thus there is one equivalence class, corresponding to a scalar particle. At the next level there are D states |e; k = e · α−1 |0; k .

(4.1.12)

e; k|e; k = 0; k|e∗ · α1 e · α−1 |0; k

= 0; k|(e∗ · e + e∗ · α−1 e · α1 )|0; k

= eµ∗ eµ (2π)D δ D (k − k ) .

(4.1.13)

µ αµ† n = α−n ,

(4.1.14)

The norm is

We have used as follows from the Hermiticity of X µ , and also

0; k|0; k = (2π)D δ D (k − k ) ,

(4.1.15)

as follows from momentum conservation. The timelike excitation has a negative norm. The Lm 0 condition gives 1+A . α The other nontrivial physical state condition is m2 =

Lm 1 |e; k ∝ p · α1 e · α−1 |0; k = e · k|0; k = 0

(4.1.16)

(4.1.17)

124

4 The string spectrum

and so k · e = 0. There is a spurious state at this level: 1/2 Lm k · α−1 |0; k . −1 |0; k = (2α )

(4.1.18)

That is, eµ ∝ k µ is spurious. There are now three cases: (i) If A > −1, the mass-squared is positive. Going to the rest frame, the physical state condition removes the negative norm timelike polarization. The spurious state (4.1.18) is not physical, k · k = 0, so there are no null states and the spectrum consists of the D − 1 positive-norm states of a massive vector particle. (ii) If A = −1, the mass-squared is zero. Now k · k = 0 so the spurious state is physical and null. Thus, (4.1.19) k · e = 0 , eµ ∼ = eµ + γkµ . This describes the D − 2 positive-norm states of a massless vector particle. (iii) If A < −1, the mass-squared is negative. The momentum is spacelike, so the physical state condition removes a positive-norm spacelike polarization. The spurious state is not physical, so we are left with a tachyonic vector particle with D − 2 positive-norm states and one negative-norm state. Case (iii) is unacceptable. Case (ii) is the same as the light-cone quantization. Case (i) is not the same as the light-cone spectrum, having diﬀerent masses and an extra state at the ﬁrst level, but does not have any obvious inconsistency. The diﬃculty is that there is no known way to give such a theory consistent interactions. Case (ii) is the one of interest in string theory, and the one we will recover from the more general approach of the next section. The result at the next level is quite interesting: it depends on the constant A and also on the spacetime dimension D. Restricting to the value A = −1 found at the ﬁrst excited level, the spectrum only agrees with the light-cone spectrum if in addition D = 26. If D > 26, there are negative-norm states; if D < 26 the OCQ spectrum has positive norm but more states than in light-cone quantization. The derivation is left to the reader. The OCQ at A = −1 and D = 26 is in fact the same as light-cone quantization at all mass levels, HOCQ = Hlight-cone ,

(4.1.20)

as will be shown in section 4.4. It is only in this case that consistent interactions are known. The extension to the closed string is straightforward. There are two sets of oscillators and two sets of Virasoro generators, so at each level the

4.1 Old covariant quantization

125

spectrum is the product of two copies of the open string spectrum (except for the normalization of p2 in Lm 0 ). The ﬁrst two levels are then 4 ; α eµν αµ−1 ˜ αν−1 |0; k , m2 = 0 , k µ eµν = k ν eµν = 0 , eµν ∼ = eµν + aµ kν + kµ bν , a · k = b · k = 0 . |0; k ,

m2 = −

(4.1.21a) (4.1.21b) (4.1.21c)

The relevant values are again A = −1, D = 26. As in light-cone quantization, there are (D − 2)2 massless states forming a traceless symmetric tensor, an antisymmetric tensor, and a scalar.

Mnemonic For more general string theories, it is useful to have a quick mnemonic for obtaining the zero-point constant. As will be derived in the following sections, the Lm 0 condition can be understood as g (Lm 0 + L0 )|ψ, ↓ = 0 .

(4.1.22)

That is, one includes the ghost contribution with the ghosts in their ground state | ↓ , which has Lg0 = −1. The L0 generators diﬀer from the Hamiltonian by a shift (2.6.10) proportional to the central charge, but the total central charge is zero in string theory so we can just as well write this condition as (H m + H g )|ψ, ↓ = 0 .

(4.1.23)

Now apply the mnemonic for zero-point energies given at the end of chapter 2. In particular, the ghosts always cancel the µ = 0, 1 oscillators because they have the same periodicities but opposite statistics. So the rule is that A is given as the sum of the zero-point energies of the transverse oscillators. This is the same rule as in the light-cone, giving 1 ) = −1. here A = 24(− 24 Incidentally, for the purposes of counting the physical states (but not for their precise form) one can simply ignore the ghost and µ = 0, 1 oscillators and count transverse excitations as in the light-cone. The condition (4.1.22) requires the weight of the matter state to be 1. Since the physical state condition requires the matter state to be a highest weight state, the vertex operator must then be a weight 1 or (1,1) tensor. This agrees with the condition from section 3.6 that the integrated vertex operator be conformally invariant, which gives another understanding of the condition A = −1.

126

4 The string spectrum 4.2

BRST quantization

We now return to a more systematic study of the spectrum. The condition (4.1.2) is not suﬃcient to guarantee gauge invariance. It implies invariance for arbitrary ﬁxed choices of gab , but this is not the most general gauge. In light-cone gauge, for example, we placed some conditions on X µ and some conditions on gab . To consider the most general possible variation of the gauge condition, we must allow δgab to be an operator, that is, to depend on the ﬁelds in the path integral. In order to derive the full invariance condition, it is useful to take a more general and abstract point of view. Consider a path integral with a local symmetry. The path integral ﬁelds are denoted φi , which in the present case would be X µ (σ) and gab (σ). Here we use a very condensed notation, where i labels the ﬁeld and also represents the coordinate σ. The gauge invariance is α δα , where again α includes the coordinate. By assumption the gauge parameters α are real, since we can always separate a complex parameter into its real and imaginary parts. The gauge transformations satisfy an algebra (4.2.1) [δα , δβ ] = f γαβ δγ . Now ﬁx the gauge by conditions F A (φ) = 0 ,

(4.2.2)

where once again A includes the coordinate. Following the same Faddeev– Popov procedure as in section 3.3, the path integral becomes

[dφi ] exp(−S1 ) → Vgauge

[dφi dBA dbA dcα ] exp(−S1 − S2 − S3 ) ,

(4.2.3)

where S1 is the original gauge-invariant action, S2 is the gauge-ﬁxing action S2 = −iBA F A (φ) ,

(4.2.4)

and S3 is the Faddeev–Popov action S3 = bA cα δα F A (φ) .

(4.2.5)

We have introduced the ﬁeld BA to produce an integral representation of the gauge-ﬁxing δ(F A ). There are two things to notice about this action. The ﬁrst is that it is invariant under the Becchi–Rouet–Stora–Tyutin (BRST) transformation δB φi = −icα δα φi , δB BA = 0 , δB bA = BA , i δB cα = f αβγ cβ cγ . 2

(4.2.6a) (4.2.6b) (4.2.6c) (4.2.6d)

4.2 BRST quantization

127

This transformation mixes commuting and anticommuting objects, so that must be taken to be anticommuting. There is a conserved ghost number, which is +1 for cα , −1 for bA and , and 0 for all other ﬁelds. The original action S1 is invariant by itself, because the action of δB on φi is just a gauge transformation with parameter icα . The variation of S2 cancels the variation of bA in S3 , while the variations of δα F A and cα in S3 cancel. The second key property is that δB (bA F A ) = i(S2 + S3 ) .

(4.2.7)

Now consider a small change δF in the gauge-ﬁxing condition. The change in the gauge-ﬁxing and ghost actions gives δ f|i = i f|δB (bA δF A )|i

= − f| { QB , bA δF A } |i ,

(4.2.8)

where we have written the BRST variation as an anticommutator with the corresponding conserved charge QB . Therefore, physical states must satisfy

ψ|{ QB , bA δF A }|ψ = 0 .

(4.2.9)

In order for this to hold for arbitrary δF, it must be that QB |ψ = QB |ψ = 0 .

(4.2.10)

This is the essential condition: physical states must be BRST-invariant. We have assumed that Q†B = QB . There are several ways to see that this must indeed be the case. One is that if Q†B were diﬀerent, it would have to be some other symmetry, and there is no candidate. A better argument is that the ﬁelds cα and bA are like anticommuting versions of the gauge parameter α and Lagrange multiplier BA , and so inherit their reality properties. As an aside, we may also add to the action a term proportional to −1 δB (bA BB M AB ) = −BA BB M AB

(4.2.11)

M AB .

By the above argument, amplitudes between for any constant matrix physical states are unaﬀected. The integral over BA now produces a Gaussian rather than a delta function: these are the Gaussian-averaged gauges, which include for example the covariant α-gauges of gauge theory. There is one more key idea. In order to move around in the space of gauge choices, the BRST charge must remain conserved. Thus it must commute with the change in the Hamiltonian, 0 = [ QB , { QB , bA δF A } ] = Q2B bA δF A − QB bA δF A QB + QB bA δF A QB − bA δF A Q2B = [ Q2B , bA δF A ] .

(4.2.12)

128

4 The string spectrum

In order for this to vanish for general changes of gauge, we need Q2B = 0 .

(4.2.13) Q2B

= constant is That is, the BRST charge is nilpotent; the possibility 2 excluded because QB has ghost number 2. The reader can check that acting twice with the BRST transformation (4.2.6) leaves all ﬁelds invariant. In particular, 1 (4.2.14) δB (δB cα ) = − f αβγ f γδ cβ cδ c = 0 . 2 The product of ghosts is antisymmetric on the indices β, δ, , and the product of structure functions then vanishes by the Jacobi identity. We should mention that we have made two simplifying assumptions about the gauge algebra (4.2.1). The ﬁrst is that the structure constants f αβγ are constants, independent of the ﬁelds, and the second is that the algebra does not have, on the right-hand side, additional terms proportional to the equations of motion. More generally, both of these assumptions break down. In these cases, the BRST formalism as we have described it does not give a nilpotent transformation, and a generalization, the Batalin– Vilkovisky (BV) formalism, is needed. The BV formalism has had various applications in string theory, but we will not have a need for it. The nilpotence of QB has an important consequence. A state of the form QB |χ

(4.2.15)

will be annihilated by QB for any χ and so is physical. However, it is orthogonal to all physical states including itself:

ψ| (QB |χ ) = ( ψ|QB ) |χ = 0

(4.2.16)

if QB |ψ = 0. All physical amplitudes involving such a null state thus vanish. Two physical states that diﬀer by a null state, |ψ = |ψ + QB |χ

(4.2.17)

will have the same inner products with all physical states and are therefore physically equivalent. So, just as for the OCQ, we identify the true physical space with a set of equivalence classes, states diﬀering by a null state being equivalent. This is a natural construction for a nilpotent operator, and is known as the cohomology of QB . Other examples of nilpotent operators are the exterior derivative in diﬀerential geometry and the boundary operator in topology. In cohomology, the term closed is often used for states annihilated by QB and the term exact for states of the form (4.2.15). Thus, our prescription is Hclosed . (4.2.18) HBRST = Hexact

129

4.2 BRST quantization

We will see explicitly in the remainder of the chapter that this space has the expected form in string theory. Essentially, the invariance condition removes one set of unphysical X µ oscillators and one set of ghost oscillators, and the equivalence relation removes the other set of unphysical X µ oscillators and the other set of ghost oscillators. Point-particle example Let us consider an example, the point particle. Expanding out the condensed notation above, the local symmetry is coordinate reparameterization δτ(τ), so the index α just becomes τ and a basis of inﬁnitesimal transformations is δτ1 τ(τ) = δ(τ − τ1 ). These act on the ﬁelds as δτ1 X µ (τ) = −δ(τ − τ1 )∂τ X µ (τ) ,

δτ1 e(τ) = −∂τ [δ(τ − τ1 )e(τ)] . (4.2.19)

Acting with a second transformation and forming the commutator, we have [δτ1 , δτ2 ]X µ (τ)

= − δ(τ − τ1 )∂τ δ(τ − τ2 ) − δ(τ − τ2 )∂τ δ(τ − τ1 ) ∂τ X µ (τ) ≡

dτ3 f τ3τ1 τ2 δτ3 X µ (τ) .

(4.2.20)

From the commutator we have determined the structure function f τ3τ1 τ2 = δ(τ3 − τ1 )∂τ3 δ(τ3 − τ2 ) − δ(τ3 − τ2 )∂τ3 δ(τ3 − τ1 ) .

(4.2.21)

The BRST transformation is then δB X µ δB e δB B δB b δB c

= = = = =

˙µ , icX ˙ , i(ce)

(4.2.22a) (4.2.22b) (4.2.22c) (4.2.22d) (4.2.22e)

0, B , ic˙c .

The gauge e(τ) = 1 is analogous to the unit gauge for the string, using the single coordinate freedom to ﬁx the single component of the tetrad, so F(τ) = 1 − e(τ). The gauge-ﬁxed action is then

S=

dτ

1 −1 ˙ µ ˙ 1 e X Xµ + em2 + iB(e − 1) − e˙bc 2 2

.

(4.2.23)

We will ﬁnd it convenient to integrate B out, thus ﬁxing e = 1. This leaves only the ﬁelds X µ , b, and c, with action

S=

1 ˙µ˙ 1 dτ X Xµ + m2 − ˙bc 2 2

(4.2.24)

130

4 The string spectrum

and BRST transformation ˙µ , δB X µ = icX 1 ˙µ˙ 1 δB b = i − X Xµ + m2 − ˙bc , 2 2 δB c = ic˙c .

(4.2.25a) (4.2.25b) (4.2.25c)

Since B is no longer present, we have used the equation of motion from e to replace B in the transformation law for b. The reader can check that the new transformation (4.2.25) is a symmetry of the action and is nilpotent. This always works when the ﬁelds B A are integrated out, though δB δB b is no longer identically zero but is proportional to the equations of motion. This is perfectly satisfactory: Q2B = 0 holds as an operator equation, which is what we need. The canonical commutators are [pµ , X ν ] = −iη µν ,

{b, c} = 1 ,

(4.2.26)

˙ µ , the i being from the Euclidean signature. The Hamiltonian with pµ = iX is H = 12 (p2 + m2 ), and the Noether procedure gives the BRST operator QB = cH .

(4.2.27)

The structure here is analogous to what we will ﬁnd for the string. The constraint (the missing equation of motion) is H = 0, and the BRST operator is c times this. The ghosts generate a two-state system, so a complete set of states is |k, ↓ , |k, ↑ , where pµ |k, ↓ = k µ |k, ↓ , pµ |k, ↑ = k µ |k, ↑ , b|k, ↓ = 0 , b|k, ↑ = |k, ↓ , c|k, ↓ = |k, ↑ , c|k, ↑ = 0 .

(4.2.28a) (4.2.28b) (4.2.28c)

The action of the BRST operator on these is QB |k, ↓ = 12 (k 2 + m2 )|k, ↑ ,

QB |k, ↑ = 0 .

(4.2.29)

From this it follows that the closed states are |k, ↓ , |k, ↑ ,

k 2 + m2 = 0 , all k µ ,

(4.2.30a) (4.2.30b)

and that the exact states are |k, ↑ ,

k 2 + m2 = 0 .

(4.2.31)

The closed states that are not exact are |k, ↓ , k 2 + m2 = 0 ;

|k, ↑ , k 2 + m2 = 0 .

(4.2.32)

4.3 BRST quantization of the string

131

That is, physical states must satisfy the mass-shell condition, but we have two copies of the expected spectrum. In fact, only states |k, ↓ satisfying the additional condition b|ψ = 0

(4.2.33)

appear in physical amplitudes. The origin of this additional condition is kinematic. For k 2 +m2 = 0, the states |k, ↑ are exact — they are orthogonal to all physical states and their amplitudes must vanish identically. So the amplitudes can only be proportional to δ(k 2 + m2 ). But amplitudes in ﬁeld theory and in string theory, while they may have poles and cuts, never have delta functions (except in D = 2 where the kinematics is special). So they must vanish identically. 4.3

BRST quantization of the string

In string theory, the BRST transformation is ¯ µ, δB X µ = i(c∂ + ˜c∂)X ˜X + T ˜ g) , δB b = i(T X + T g ) , δB ˜b = i(T ¯c . δB c = ic∂c , δB˜c = i˜c∂˜

(4.3.1a) (4.3.1b) (4.3.1c)

The reader can derive this following the point-particle example. To the sum of the Polyakov and ghost actions, add the gauge-ﬁxing term

i d2 σ g 1/2 B ab (δab − gab ) . (4.3.2) 4π The transformation δB bab = Bab becomes (4.3.1) after integrating over Bab and using the gab equation of motion to replace Bab in the transformation. The Weyl ghost is just a Lagrange multiplier, making bab traceless. Again, the reader can check nilpotence up to the equations of motion. The similarity between the string BRST transformation (4.3.1) and the particle case (4.2.25) is evident. Replacing the X µ with a general matter CFT, the BRST transformation of the matter ﬁelds is a conformal transformation with v(z) = c(z), while T m replaces T X in the transformation of b. Noether’s theorem gives the BRST current 1 3 : cT g : + ∂2 c , 2 2 3 = cT m + : bc∂c : + ∂2 c , (4.3.3) 2 and correspondingly for ˜B . The ﬁnal term in the current is a total derivative and does not contribute to the BRST charge; it has been added by hand to make the BRST current a tensor. The OPEs of the BRST jB = cT m +

132

4 The string spectrum

current with the ghost ﬁelds and with a general matter tensor ﬁeld are 3 1 1 + 2 j g (0) + T m+g (0) , (4.3.4a) 3 z z z 1 jB (z)c(0) ∼ c∂c(0) , (4.3.4b) z & h 1% jB (z)Om (0, 0) ∼ 2 cOm (0, 0) + h(∂c)Om (0, 0) + c∂Om (0, 0) . (4.3.4c) z z The simple poles reﬂect the BRST transformations (4.3.1) of these ﬁelds. The BRST operator is jB (z)b(0) ∼

1 QB = 2πi

(dz jB − d¯z ˜B ) .

(4.3.5)

By the usual contour argument, the OPE implies g {QB , bm } = Lm m + Lm .

(4.3.6)

In terms of the ghost modes, QB =

∞

˜m (cn Lm cn L −n + ˜ −n )

n=−∞

+

∞ (m − n) ◦

m,n=−∞

2

◦

(cm cn b−m−n + ˜cm˜cn ˜b−m−n )◦◦ + aB (c0 + ˜c0 ) . (4.3.7)

The ordering constant is aB = ag = −1, as follows from the anticommutator g {QB , b0 } = Lm 0 + L0 .

(4.3.8)

There is an anomaly in the gauge symmetry when cm = 26, so we expect some breakdown in the BRST formalism. The BRST current is still conserved: all of the terms in the current (4.3.3) are analytic for any value of the central charge. However, it is no longer nilpotent, {QB , QB } = 0 only if cm = 26 .

(4.3.9)

The shortest derivation of this uses the Jacobi identity, exercise 4.3. More directly it follows from the OPE jB (z)jB (0) ∼ −

cm − 18 cm − 18 2 cm − 26 3 c∂c(0)− c∂ c(0)− c∂ c(0) . (4.3.10) 2z 3 4z 2 12z

This requires a bit of calculation, being careful about minus signs from anticommutation. The single pole implies that {QB , QB } = 0 when cm = 26.

133

4.3 BRST quantization of the string Note also the OPE

cm − 26 1 1 c(0) + 2 jB (0) + ∂jB (0) , (4.3.11) 4 2z z z which implies that jB is a tensor only when cm = 26. Let us point out some important features. The missing equation of motion is just the vanishing of the generator Tab of the conformal transformations, the part of the local symmetry group not ﬁxed by the gauge choice. This is a general result. When the gauge conditions ﬁx the symmetry completely, one can show that the missing equations of motion are trivial due to the gauge invariance of the action. When they do not, so that there is a residual symmetry group, the missing equations of motion require its generators to vanish. They must vanish in the sense (4.1.2) of matrix elements, or more generally in the BRST sense. Denote the generators of the residual symmetry group by GI . The GI are called constraints; ˜ m . They form an algebra for the string these are Lm and L T (z)jB (0) ∼

[GI , GJ ] = ig KIJ GK .

(4.3.12)

Associated with each generator is a pair of ghosts, bI and cI , where {cI , bJ } = δ IJ ,

{cI , cJ } = {bI , bJ } = 0 .

(4.3.13)

The general form of the BRST operator, as illustrated by the string case (4.3.7), is i K I J QB = cI Gm I − g IJ c c bK 2 1 g I m = c GI + GI , 2

(4.3.14)

where Gm I is the matter part of GI and GgI = −ig KIJ cJ bK Gm I

(4.3.15)

GgI

is the ghost part. The and satisfy the same algebra (4.3.12) as the GI . Using the commutators (4.3.12) and (4.3.13), one ﬁnds 1 1 (4.3.16) Q2B = {QB , QB } = − g KIJ g MKL cI cJ cL bM = 0 . 2 2 The last equality follows from the GGG Jacobi identity, which requires g KIJ g MKL to vanish when antisymmetrized on IJL. We have neglected central charge terms; these need to be checked by hand. To reiterate, a constraint algebra is a world-sheet symmetry algebra that is required to vanish in physical matrix elements. When we go on to generalize the bosonic string theory in volume two, it will be easiest to do so directly in terms of the constraint algebra, and we will write the BRST charge directly in the form (4.3.14).

134

4 The string spectrum BRST cohomology of the string

Let us now look at the BRST cohomology at the lowest levels of the string. The inner product is deﬁned by specifying (αµm )† = αµ−m , (bm )† = b−m , (cm )† = c−m ,

(˜αµm )† = α˜µ−m , (˜bm )† = ˜b−m , (˜cm )† = ˜c−m .

(4.3.17a) (4.3.17b) (4.3.17c)

In particular, the Hermiticity of the BRST charge requires that the ghost ﬁelds be Hermitean as well. The Hermiticity of the ghost zero modes forces the inner products of the ground state to take the form open string: closed string:

(4.3.18a)

0; k|c0 |0; k = (2π)26 δ 26 (k − k ) , 26 26

0; k|˜c0 c0 |0; k = i(2π) δ (k − k ) . (4.3.18b)

Here |0; k denotes the matter ground state times the ghost ground state | ↓ , with momentum k. The c0 and ˜c0 insertions are necessary for a nonzero result. For example, 0; k|0; k = 0; k|(c0 b0 + b0 c0 )|0; k = 0; the last equality follows because b0 annihilates both the bra and the ket. The factor of i is needed in the ghost zero-mode inner product for Hermiticity. Inner products of general states are then obtained, as in the earlier calculation (4.1.13), by use of the commutation relations and the adjoints (4.3.17). We will focus on the open string; the closed string discussion is entirely parallel but requires twice as much writing. As in the point-particle case (4.2.33), we will assert (and later show) that physical states must satisfy the additional condition b0 |ψ = 0 .

(4.3.19)

L0 |ψ = {QB , b0 }|ψ = 0 ,

(4.3.20)

This also implies since QB and b0 both annihilate |ψ . The operator L0 is L0 = α (pµ pµ + m2 ) , where α m2 =

∞ n=1

n Nbn + Ncn +

25

(4.3.21)

Nµn

− 1.

(4.3.22)

µ=0

Thus the L0 condition (4.3.20) determines the mass of the string. BRST invariance, with the extra condition (4.3.19), implies that every string state ˆ the space of states satisfying is on the mass shell. We will denote by H the conditions (4.3.19) and (4.3.20). From the commutators {QB , b0 } = L0 ˆ into itself. and [QB , L0 ] = 0, it follows that QB takes H

4.3 BRST quantization of the string

135

ˆ It The inner product (4.3.18a) is not quite the correct one to use in H. 26 is not well deﬁned: the ghost zero modes give zero, while the δ (k − k ) contains a factor δ(0) because the momentum support is restricted to the ˆ a reduced inner product in which mass shell. Therefore we use in H 0 we simply ignore the X and ghost zero modes. It is this inner product that will be relevant for the probabilistic interpretation. One can check ˆ that QB is still Hermitean with the reduced inner product, in the space H. 0 Note that the mass-shell condition determines k in terms of the spatial momentum k (we focus on the incoming case, k 0 > 0), and that we have used covariant normalization for states. Let us now work out the ﬁrst levels of the D = 26 ﬂat spacetime string. At the lowest level, N = 0, we have |0; k ,

−k 2 = −

1 . α

(4.3.23)

This state is invariant, QB |0; k = 0 ,

(4.3.24)

because every term in QB contains either lowering operators or L0 . This also shows that there are no exact states at this level, so each invariant state corresponds to a cohomology class. These are just the states of the tachyon. The mass-shell condition is the same as found in the light-cone quantization of section 1.3 and from the open string vertex operators in section 3.6. The tachyon mass-squared, determined in a heuristic way in chapter 1, is here ﬁxed by the normal ordering constant in QB , which is determined by the requirement of nilpotence. At the next level, N = 1, there are 26 + 2 states, |ψ1 = (e · α−1 + βb−1 + γc−1 )|0; k ,

−k 2 = 0 ,

(4.3.25)

depending on a 26-vector eµ and two constants, β and γ. The norm of this state is

ψ1 ψ1 = 0; k(e* · α1 + β*b1 + γ*c1 )(e · α−1 + βb−1 + γc−1 )|0; k

= (e* · e + β*γ + γ*β) 0; k0; k . (4.3.26) Going to an orthogonal basis, there are 26 positive-norm states and 2 negative-norm states. The BRST condition is 0 = QB |ψ1 = (2α )1/2 (c−1 k · α1 + c1 k · α−1 )|ψ1

= (2α )1/2 (k · ec−1 + βk · α−1 )|0; k .

(4.3.27)

The terms proportional to c0 sum to zero by the mass-shell condition, and are omitted. An invariant state therefore satisﬁes k · e = β = 0. There are

136

4 The string spectrum

26 linearly independent states remaining, of which 24 have positive norm and 2 have zero norm, being orthogonal to all physical states including themselves. The zero-norm invariant states are created by c−1 and k · α−1 . A general |χ is of the same form (4.3.25) with constants eµ , β , γ , so the general BRST-exact state at this level is QB |χ = (2α )1/2 (k · e c−1 + β k · α−1 )|0; k .

(4.3.28)

Thus the ghost state c−1 |0; k is BRST-exact, while the polarization is transverse with the equivalence relation eµ ∼ = eµ + (2α )1/2 β kµ . This leaves the 24 positive-norm states expected for a massless vector particle, in agreement with the light-cone quantization and with the OCQ at A = −1. This pattern is general: there are two extra positive-norm and two extra negative-norm families of oscillators, as compared to the lightcone quantization. The physical state condition eliminates two of these and leaves two combinations with vanishing inner products. These null oscillators are BRST-exact and are removed by the equivalence relation. An aside: the states b−1 |0; k and c−1 |0; k can be identiﬁed with the two Faddeev–Popov ghosts that appear in the BRST-invariant quantization of a massless vector ﬁeld in ﬁeld theory. The world-sheet BRST operator acts on these states in the same way as the corresponding spacetime BRST operator in gauge ﬁeld theory. Acting on the full string Hilbert space, the open string BRST operator is then some inﬁnite-dimensional generalization of the spacetime gauge theory BRST invariance, while the closed string BRST operator is a generalization of the spacetime general coordinate BRST invariance, in the free limit. The generalization to the closed string is straightforward. We restrict ˆ of states satisfying1 attention to the space H b0 |ψ = ˜b0 |ψ = 0 ,

(4.3.29)

˜ 0 |ψ = 0 . L0 |ψ = L

(4.3.30)

implying also

In the closed string, L0 =

1

α 2 (p + m2 ) , 4

˜ 0 = α (p2 + m ˜ 2) , L 4

(4.3.31)

S. Weinberg has pointed out that the simple argument given at the end of section 4.2 would only lead to the weaker condition (b0 + ˜ b0 )|ψ = 0. Nevertheless, we shall see from the detailed form of string amplitudes in chapter 9 that there is a projection onto states satisfying conditions (4.3.29).

137

4.4 The no-ghost theorem where

∞ 25 α 2 m = n Nbn + Ncn + Nµn 4 n=1 µ=0

∞ 25 α 2 ˜ bn + N ˜ µn ˜ cn + ˜ = N n N m 4 n=1 µ=0

− 1,

(4.3.32a)

− 1.

(4.3.32b)

Repeating the earlier exercise, we ﬁnd at m2 = −4/α the tachyon, and at m2 = 0 the 24 × 24 states of the graviton, dilaton, and antisymmetric tensor.

4.4

The no-ghost theorem

In this section we prove that the BRST cohomology of the string has a positive inner product and is isomorphic to the light-cone and OCQ spectra. We are identifying the BRST cohomology as the physical Hilbert space. We will also need to verify, in our study of string amplitudes, that the amplitudes are well deﬁned on the cohomology (that is, that equivalent states have equal amplitudes) and that the S-matrix is unitary within the space of physical states. We will work in the general framework described at the end of chapter 3. That is, the world-sheet theory consists of d free X µ ﬁelds including µ = 0, plus some compact unitary CFT K of central charge 26 − d, plus the ghosts. The Virasoro generators are a sum K g Lm = LX m + Lm + Lm .

(4.4.1)

By compact we mean that LK 0 has a discrete spectrum. For example, in the case that K corresponds to strings on a compact manifold, the term α p2 in L0 is replaced by the Laplacian −α ∇2 , which has a discrete spectrum on a compact space. The general state is labeled |N, I; k ,

˜ I; k , |N, N,

(4.4.2)

˜ refer to both in the open and closed strings respectively, where N (and N) the d-dimensional and ghost oscillators, k is the d-momentum, and I labels the states of the compact CFT with given boundary conditions. The b0 condition (4.3.19) or (4.3.29) is imposed as above, implying the mass-shell

138

4 The string spectrum

condition −

d−1

kµ k µ = m2 ,

µ=0

2

αm =

∞

(4.4.3a)

n Nbn + Ncn +

n=1

d−1

+ LK 0 − 1 ,

Nµn

(4.4.3b)

µ=0

for the open string and −

d−1 µ=0

˜2 , kµ k µ = m2 = m

(4.4.4a)

∞ d−1 α 2 n Nbn + Ncn + Nµn m = 4 n=1 µ=0

∞ d−1 α 2 ˜ bn + N ˜ µn ˜ cn + ˜ = N n N m 4 n=1 µ=0

+ LK 0 − 1 ,

(4.4.4b)

˜K + L 0 − 1 ,

(4.4.4c)

for the closed string. That is, the contributions of the d ≤ µ ≤ 25 oscillators ˜K are replaced by the eigenvalue of LK 0 or L0 from the compact CFT. The only information we use about the compact CFT is that it is conformally invariant with the appropriate central charge, so there is a nilpotent BRST operator, and that it has a positive inner product. The basis I can be taken to be orthonormal, so the reduced inner products are

0, I; k0, I ; k = 0, 0, I; k0, 0, I ; k = 2k 0 (2π)d−1 δ d−1 (k − k )δI,I . (4.4.5) Now let us see what we expect for the physical Hilbert space. Deﬁne ˆ that have the transverse Hilbert space H⊥ to consist of those states in H 0 1 no longitudinal (X , X , b, or c) excitations. Since these oscillators are the source of the indeﬁnite inner product, H⊥ has a positive inner product. Light-cone gauge-ﬁxing eliminates the longitudinal oscillators directly — the light-cone Hilbert space is isomorphic to H⊥ , as one sees explicitly in the ﬂat-spacetime case from chapter 1. We will show, in the general case, that the BRST cohomology is isomorphic to H⊥ . That is, it has the same number of states at each mass level, and has a positive inner product. This is the no-ghost theorem. Proof The proof has two parts. The ﬁrst is to ﬁnd the cohomology of a simpliﬁed BRST operator Q1 , which is quadratic in the oscillators, and the second is to show that the cohomology of the full QB is identical to that of Q1 .

139

4.4 The no-ghost theorem Deﬁne the light-cone oscillators −1/2 0 α± (αm ± α1m ) , m =2

(4.4.6)

which satisfy − [α+ m , αn ] = −mδm,−n ,

+ − − [α+ m , αn ] = [αm , αn ] = 0 .

(4.4.7)

We will use extensively the quantum number N lc =

∞ 1 m=−∞ m=0

m

◦ ◦

− α+ −m αm

◦ ◦

(4.4.8)

.

The number N lc counts the number of − excitations minus the number of + excitations; it is not a Lorentz generator because the center-ofmass piece has been omitted. We choose a Lorentz frame in which the momentum component k + is nonzero. Now decompose the BRST generator using the quantum number N lc : QB = Q1 + Q0 + Q−1 ,

(4.4.9)

where Qj changes N lc by j units, [N lc , Qj ] = jQj .

(4.4.10)

Also, each of the Qj increases the ghost number N g by one unit: [N g , Qj ] = Qj . Expanding out Q2B = 0 gives

(4.4.11)

Q21 + {Q1 , Q0 } + {Q1 , Q−1 }+Q20 + {Q0 , Q−1 } + Q2−1 = 0 . (4.4.12)

Each group in parentheses has a diﬀerent N lc and so must vanish separately. In particular, Q1 itself is nilpotent, and so has a cohomology. Explicitly, Q1 = −(2α )1/2 k +

∞

α− −m cm .

(4.4.13)

m=−∞ m=0

For m < 0 this destroys a + mode and creates a c, and for m > 0 it creates a − mode and destroys a b. One can ﬁnd the cohomology directly by considering the action of Q1 in the occupation basis. We will leave this as an exercise, and instead use a standard strategy which will also be useful in generalizing to QB . Deﬁne ∞ 1 α+ b R= 1/2 + (2α ) k m=−∞ −m m m=0

(4.4.14)

140

4 The string spectrum

and S ≡ {Q1 , R} = =

∞ ' m=1 ∞

− − + mb−m cm + mc−m bm − α+ −m αm − α−m αm

m(Nbm + Ncm + Nm+ + Nm− ) .

(

(4.4.15)

m=1

The normal ordering constant is determined by noting that Q1 and R both annihilate the ground state. Note that Q1 commutes with S . We can then calculate the cohomology within each eigenspace of S , and the full cohomology is the union of the results. If |ψ is Q1 -invariant with S |ψ = s|ψ , then for nonzero s 1 1 |ψ = {Q1 , R}|ψ = Q1 R|ψ , s s

(4.4.16)

and so |ψ is actually Q1 -exact. Therefore the Q1 cohomology can be nonzero only at s = 0. By the deﬁnition (4.4.15) of S , the s = 0 states have no longitudinal excitations — the s = 0 space is just H⊥ . The operator Q1 annihilates all states in H⊥ , so they are all Q1 -closed and there are no Q1 -exact states in this space. Therefore the cohomology is H⊥ itself. We have proven the no-ghost theorem, but for the operator Q1 , not QB . The proof had two steps, the ﬁrst being to show that the cohomology could only come from s = 0 states (the kernel of S ), and the second to show that s = 0 states were Q1 -invariant. It is useful to prove the second step in a more abstract way, using the property that all s = 0 states have the same ghost number, in this case − 12 . Suppose S |ψ = 0. Since S and Q1 commute we have 0 = Q1 S |ψ = S Q1 |ψ .

(4.4.17)

The state |ψ has ghost number − 12 , so Q1 |ψ has ghost number + 12 . Since S is invertible at this ghost number, it must be that Q1 |ψ = 0 as we wished to show. It remains to show that the cohomology of QB is the same as the cohomology of Q1 . The idea here is to use in place of S the operator S + U ≡ {QB , R} .

(4.4.18)

Now, U = {Q0 + Q−1 , R} lowers N lc by one or two units. In terms of N lc y, S is diagonal and U is lower triangular. By general properties of lower triangular matrices, the kernel of S + U can be no larger than the kernel of its diagonal part S . In fact they are isomorphic: if |ψ0 is annihilated by S , then |ψ = (1 − S −1 U + S −1 US −1 U − . . .)|ψ0

(4.4.19)

4.4 The no-ghost theorem

141

is annihilated by S + U. The factors of S −1 make sense because they always act on states of N lc < 0, where S is invertible. For the same reason, S + U is invertible except at ghost number − 12 . Eqs. (4.4.16) and (4.4.17) can now be applied to QB , with S + U replacing S . They imply that the QB cohomology is isomorphic to the kernel of S + U, which is isomorphic to the kernel of S , which is isomorphic to the cohomology of Q1 , as we wished to show. We must still check that the inner product is positive. All terms after the ﬁrst on the right-hand side of eq. (4.4.19) have strictly negative N lc . By the commutation relations, the inner product is nonzero only between states whose N lc adds to zero. Then for two states (4.4.19) in the kernel of S + U,

ψψ = ψ0 ψ0 .

(4.4.20)

The positivity of the inner product on the kernel of S + U then follows from that for the kernel of S , and we are done. After adding the tilded operators to QB , N lc , R, etc., the closed string proof is identical. BRST–OCQ equivalence We now prove the equivalence HOCQ = HBRST = Hlight-cone .

(4.4.21)

To a state |ψ in the matter Hilbert space, associate the state |ψ, ↓

(4.4.22)

from the full matter plus ghost theory. Again the ghost vacuum | ↓ is annihilated by all of the ghost lowering operators, bn for n ≥ 0 and cn for n > 0. Acting with QB , QB |ψ, ↓ =

∞

c−n (Lm n − δn,0 )|ψ, ↓ = 0 .

(4.4.23)

n=0

All terms in QB that contain ghost lowering operators drop out, and the constant in the n = 0 term is known from the mode expansion of QB , eq. (4.3.7). Each OCQ physical state thus maps to a BRST-closed state. The ordering constant A = −1 arises here from the L0 eigenvalue of the ghost vacuum. To establish the equivalence (4.4.21), we need to show more. First, we need to show that we have a well-deﬁned map of equivalence classes: that if ψ and ψ are equivalent OCQ physical states, they map into the same BRST class: |ψ, ↓ − |ψ , ↓

(4.4.24)

142

4 The string spectrum

must be BRST-exact. By (4.4.23), this state is BRST-closed. Further, since |ψ − |ψ is OCQ null, the state (4.4.24) has zero norm. From the BRST no-ghost theorem, the inner product on the cohomology is positive, so a zero-norm closed state is BRST-exact, as we needed to show. To conclude that we have an isomorphism, we need to show that the map is one-to-one and onto. One-to-one means that OCQ physical states ψ and ψ that map into the same BRST class must be in the same OCQ class — if |ψ, ↓ − |ψ , ↓ = QB |χ ,

(4.4.25)

then |ψ − |ψ must be OCQ null. To see this, expand |χ =

∞

b−n |χn , ↓ + . . . ;

(4.4.26)

n=1

|χ has ghost number − 32 , so the ellipsis stands for terms with at least one c and two b excitations. Insert this into the form (4.4.25) and keep on both sides only terms with the ghost ground state. This gives |ψ, ↓ − |ψ , ↓ = =

∞

cm Lm −m b−n |χn , ↓

m,n=1 ∞

Lm −n |χn , ↓ .

(4.4.27)

n=1

Terms with ghost excitations must vanish separately, and so have been m omitted. Thus, |ψ − |ψ = ∞ n=1 L−n |χn is OCQ null, and the map is one-to-one. Finally, we must show that the map is onto, that every QB class contains at least one state of the form (4.4.22). In fact, the speciﬁc representatives (4.4.19), the states annihilated by S + U, are of this form. To see this, consider the quantum number N = 2N − + Nb + Nc , involving the total numbers of −, b, and c excitations. The operator R has N = −1: terms in R with m > 0 reduce Nc by one unit and terms with m < 0 reduce N − by one unit and increase Nb by one unit. Examining Q0 + Q−1 , one ﬁnds various terms with N = 1, but no greater. So U = {R, Q0 + Q−1 } cannot increase N . Examining the state (4.4.19), and noting that S and |ψ0 have N = 0, we see that all terms on the right-hand side have N ≤ 0. By deﬁnition N is nonnegative, so it must be that N |ψ = 0. This implies no −, b, or c excitations, and so this state is of the form (4.4.22). Thus the equivalence (4.4.21) is shown. The full power of the BRST method is needed for understanding the general structure of string amplitudes. However, for many practical purposes it is a great simpliﬁcation to work with the special states of the

Exercises

143

form |ψ, ↓ , and so it is useful to know that every BRST class contains at least one state in which the ghost modes are unexcited. We will call these OCQ-type states. The OCQ physical state condition (4.1.5) requires that the matter state be a highest weight state with L0 = 1. The corresponding vertex operator is a weight-1 tensor ﬁeld Vm constructed from the matter ﬁelds. Including the ghost state | ↓ , the full vertex operator is cVm . For the closed string the full vertex operator is ˜ccVm with Vm a (1, 1) tensor. The matter part of the vertex operator is the same as found in the Polyakov formalism in section 3.6, while the ghost part has a simple interpretation that we will encounter in the next chapter. Eq. (4.4.19) deﬁnes an OCQ-physical state in each cohomology class. For the special case of ﬂat spacetime this state can be constructed more explicitly by using the Del Giudice–Di Vecchia–Fubini (DDF) operators. To explain these we will need to develop some more vertex operator technology, so this is deferred to chapter 8. Exercises 4.1 Extend the OCQ, for A = −1 and general D, to the second excited level of the open string. Verify the assertions made in the text about extra positive- and negative-norm states. 4.2 (a) In the OCQ, show that the state Lm −1 |χ is physical (and therefore null) if |χ is a highest weight state with Lm 0 = 0. 3 m m + L L )|χ

is physical and null if |χ is a (b) Show that the state (Lm −2 2 −1 −1 = −1. highest weight state with Lm 0 4.3 By writing out terms, demonstrate the Jacobi identity { [QB , Lm ] , bn } − { [Lm , bn ] , QB } − [ {bn , QB } , Lm ] = 0 . This is the sum of cyclic permutations, with commutators or anticommutators as appropriate, and minus signs from anticommutation. Use the Jacobi identity and the (anti)commutators (2.6.24) and (4.3.6) of bn with Ln and QB to show that { [QB , Lm ] , bn } vanishes when the total central charge is zero. This implies that [QB , Lm ] contains no c modes. However, this commutator has N g = 1, and so it must vanish: the charge QB is conformally invariant. Now use the QB QB bn Jacobi identity in the same way to show that QB is nilpotent when the total central charge is zero. 4.4 In more general situations, one encounters graded Lie algebras, with bosonic generators (fermion number FI even) and fermionic generators (FI odd). The algebra of constraints is GI GJ − (−1)FI FJ GJ GI = ig KIJ GK .

144

4 The string spectrum

Construct a nilpotent BRST operator. You will need bosonic ghosts for the fermionic constraints. 4.5 (a) Carry out the BRST quantization for the ﬁrst two levels of the closed string explicitly. (b) Carry out the BRST quantization for the third level of the open string, m2 = 1/α . − 4.6 (a) Consider the operator Q = α− −1 c1 + α1 c−1 , obtained by truncating Q1 to the m = ±1 oscillators only. Calculate the cohomology of Q directly, by considering its action on a general linear combination of the states −

N N (α− |0 . (b−1 )Nb (c−1 )Nc (α+ −1 ) −1 ) +

(b) Generalize this to the full Q1 .

5 The string S-matrix

In chapter 3 we wrote the string S-matrix as a path integral over compact two-dimensional surfaces with vertex operators. In this chapter we will reduce the path integral to gauge-ﬁxed form. 5.1

The circle and the torus

We would like to identify a gauge slice, a choice of one conﬁguration from each (diﬀ×Weyl)-equivalent set as in ﬁgure 3.7. Locally we did this by ﬁxing the metric, but globally there is a small mismatch between the space of metrics and the world-sheet gauge group. Once again the point particle is a good illustration. We want to evaluate the Euclidean path integral

1 [de dX] exp − 2

−1 ˙ µ ˙

2

dτ (e X Xµ + em ) .

(5.1.1)

Consider a path forming a closed loop in spacetime, so the topology is a circle. The parameter τ can be taken to run from 0 to 1 with the endpoints identiﬁed. That is, X µ (τ) and e(τ) are periodic on 0 ≤ τ ≤ 1. The tetrad e(τ) has one component and there is one local symmetry, the choice of parameter, so as for the string there is just enough local symmetry to ﬁx the tetrad completely. The tetrad transforms as e dτ = edτ. The gauge choice e = 1 thus gives a diﬀerential equation for τ (τ), ∂τ = e(τ) . ∂τ

(5.1.2)

Integrating this with the boundary condition τ (0) = 0 determines τ (τ) =

τ

dτ e(τ ) .

0

145

(5.1.3)

146

5 The string S-matrix

The complication is that in general τ (1) = 1, so the periodicity is not preserved. In fact,

1

τ (1) =

dτ e(τ) = l

(5.1.4)

0

is the invariant length of the circle. So we cannot simultaneously set e = 1 and keep the coordinate region ﬁxed. We can hold the coordinate region ﬁxed and set e to the constant value e = l, or set e = 1 and let the coordinate region vary: e = l ,

0≤τ≤1,

(5.1.5a)

e = 1 ,

0≤τ≤l .

(5.1.5b)

or

In either case, after ﬁxing the gauge invariance we are left with an ordinary integral over l. In other words, not all tetrads on the circle are diﬀ-equivalent. There is a one-parameter family of inequivalent tetrads, parameterized by l. Both descriptions (5.1.5) will have analogs in the string. In practice, we will deﬁne the path integral using a description like (5.1.5a), where the ﬁelds are functions on a ﬁxed coordinate region, and then convert to a description like (5.1.5b) where the metric is ﬁxed and the moduli are encoded in the coordinate range. There is a second complication with the gauge-ﬁxing. The condition e = constant is preserved by the rigid translation τ → τ + v mod 1 .

(5.1.6)

That is, we have not said where on the circle to put the origin of the coordinate system. Fixing the metric thus leaves a small part of the local symmetry unﬁxed. So there is a small mismatch in both directions, metrics that are not gauge-equivalent to one another and gauge transformations that are not ﬁxed by the choice of metric. Moving on to the string we take as an example the torus, where the same complications arise. Start with the coordinate region 0 ≤ σ 1 ≤ 2π ,

0 ≤ σ 2 ≤ 2π ,

(5.1.7)

with X µ (σ 1 , σ 2 ) and gab (σ1 , σ2 ) periodic in both directions. Equivalently we can think of this as the σ plane with the identiﬁcation of points, (5.1.8) (σ 1 , σ 2 ) ∼ = (σ 1 , σ 2 ) + 2π(m, n) for integer m and n. To what extent is the ﬁeld space diﬀ×Weyl redundant? The theorem is that it is not possible to bring a general metric to unit form by a

147

5.1 The circle and the torus

2 π (τ +1) B A w 2π

0

¯ The upper and lower Fig. 5.1. Torus with modulus τ, in gauge ds2 = dwdw. edges are identiﬁed, as are the right- and left-hand edges. Closed curves A and B are marked for later reference.

diﬀ×Weyl transformation that leaves invariant the periodicity (5.1.7), but it is possible to bring it to the form ds2 = |dσ 1 + τdσ 2 |2

(5.1.9)

where τ is a complex constant. For τ = i this would be the unit metric δab . To see this, repeat the steps used in the local discussion in section 3.3. We can ﬁrst make the metric ﬂat by a Weyl transformation satisfying 2∇2 ω = R. By expanding in eigenmodes of ∇2 , one sees that with the periodic boundary conditions this has a solution up to addition unique 2 1/2 of a constant to ω. It is important here that d σ g R, which is 4π times the Euler number, vanishes for the torus. A transformation to new ˜ a then brings the metric to unit form. However, as in the coordinates σ point-particle case, there is no guarantee that this respects the original periodicity. Rather, we may now have ˜ a + 2π(mua + nv a ) ˜a ∼ σ = σ ua

(5.1.10)

va.

and By rotating and rescaling the coordiwith general translations nate system, accompanied by a shift in ω to keep the metric normalized, we can always set u = (1, 0). This leaves two parameters, the components ¯ and the periodicity is ˜ 1 + i˜ σ 2 , the metric is dwdw of v. Deﬁning w = σ w∼ = w + 2π(m + nτ) ,

(5.1.11)

where τ = v 1 + iv 2 . The torus is a parallelogram in the w-plane with periodic boundary conditions, as shown in ﬁgure 5.1. Alternatively, deﬁne σ a by w = σ 1 + τσ 2 . The original periodicity (5.1.8) is preserved but the metric now takes the more general form (5.1.9). The integration over metrics reduces to two ordinary integrals, over the real and imaginary parts of τ. The metric (5.1.9) is invariant under complex

148

5 The string S-matrix

conjugation of τ and degenerate for τ real, so we can restrict attention to Im τ > 0. As in the case (5.1.5) of the circle, we can put these parameters either in the metric (5.1.9) or the periodicity (5.1.11). The parameter τ is known as a Teichm¨uller parameter or more commonly a modulus. A diﬀerence from the point-particle case is that there is no simple invariant expression for τ analogous to the invariant length of the circle. There is some additional redundancy, which does not have an analog in the point-particle case. The value τ + 1 generates the same set of identiﬁcations (5.1.11) as τ, replacing (m, n) → (m−n, n). So also does −1/τ, deﬁning w = τw and replacing (m, n) → (n, −m). Repeated application of these two transformations, T : τ = τ + 1 ,

S : τ = −1/τ ,

(5.1.12)

generates τ =

aτ + b cτ + d

(5.1.13)

for all integer a, b, c, d such that ad − bc = 1. We can also think about this as follows. The transformation

σ1 σ2

=

d b c a

σ 1 σ 2

(5.1.14)

takes the metric (5.1.9) for σ into a metric of the same form in σ but with the modulus τ . This is a diﬀeomorphism of the torus. It is one-to-one as a consequence of ad − bc = 1 and it preserves the periodicity (5.1.8). However, it cannot be obtained from the identity by successive inﬁnitesimal transformations — it is a so-called large coordinate transformation. The curve A in the coordinate σ maps to a curve in the σ coordinate that runs a times in the A direction and −c times in the B direction. These large coordinate transformations form the group S L(2, Z), integer-valued 2 × 2 matrices with unit determinant. The group on the τ-plane is S L(2, Z)/Z2 = P S L(2, Z), because τ is unchanged if all the signs of a, b, c, d are reversed. The group of transformations (5.1.14) is known as the modular group. Using the modular transformations (5.1.13), it can be shown that every τ is equivalent to exactly one point in the region F0 shown in ﬁgure 5.2, 1 1 ≤ Re τ ≤ , |τ| ≥ 1 , (5.1.15) 2 2 except on the boundaries which are identiﬁed as shown in the ﬁgure. This is called a fundamental region for the upper half-plane mod P S L(2, Z). Because of the identiﬁcations one should think of F0 as being rolled up, and open only at Im τ → ∞. The fundamental region F0 is one representation of the moduli space of (diﬀ×Weyl)-inequivalent metrics. −

149

5.1 The circle and the torus

I

I'

i II

II'

III

III'

τ 1 − 2

1 −− 2

Fig. 5.2. The standard fundamental region F0 for the moduli space of the torus, shaded. The lines I and I are identiﬁed, as are the arcs II and II . A diﬀerent fundamental region, mapped into F0 by the modular transformation S, is bounded by II, II , III, and III .

There is a further complication as there was for the point particle. Requiring the metric to take the form (5.1.9) with τ in a given fundamental region does not ﬁx all of the diﬀ×Weyl invariance. The metric and periodicity are left invariant by rigid translations, σa → σa + va ,

(5.1.16)

so this two-parameter subgroup of diﬀ×Weyl is not ﬁxed. In addition, the discrete transformation σ a → −σ a leaves the metric invariant, and in the unoriented case also (σ 1 , σ 2 ) → (−σ 1 , σ 2 ) with τ → −¯τ. Thus there are again two kinds of mismatch between the metric degrees of freedom and the local symmetries: parameters in the metric that cannot be removed by the symmetries, and symmetries that are not ﬁxed by the choice of metric. The unﬁxed symmetries are known as the conformal Killing group (CKG). When there are enough vertex operators in the amplitude, we can fully ﬁx the gauge invariance by ﬁxing some of the vertex operator positions. On the torus with n vertex operators, the invariance (5.1.16) can be used

150

5 The string S-matrix

to ﬁx one vertex operator position, leaving an integral over τ and the n − 1 other positions. The Z2 from σ a → −σ a can be ﬁxed by restricting a second vertex operator to half the torus. Actually, for ﬁnite overcounting it is often more convenient to leave some symmetry unﬁxed and divide by an appropriate factor. Also, if there are too few vertex operators, some conformal Killing invariance remains unﬁxed and we must allow explicitly for the overcounting. A note to the reader In the remainder of this chapter we discuss moduli in more generality, and obtain the measure for the gauge-ﬁxed integral. The principal goal is the gauge-ﬁxed expression (5.3.9) for the string S-matrix. In fact, most of the interesting string physics can be seen in the tree-level and one-loop amplitudes. For these the measure can be obtained by shortcuts. For the tree-level amplitudes there are no metric moduli but some vertex operator positions must be ﬁxed. The measure can be deduced as explained below eq. (6.4.4). For one-loop amplitudes the measure can be understood in a fairly intuitive way, by analogy to the discussion of the point particle measure below eq. (7.3.9). Thus it is possible for the reader to skip the remainder of this chapter (or to skim as far as eq. (5.3.9)), and over all sections dealing with the Faddeev–Popov ghosts. 5.2

Moduli and Riemann surfaces

Now let us repeat the preceding discussion in a more general and abstract way. We start with the integral over all metrics on some given topology r. Call the space of metrics Gr . For closed oriented surfaces we can label r simply by the number g of handles, also known as the genus. After taking into account the diﬀ×Weyl redundancy, we are left with the moduli space Mr =

Gr . (diﬀ × Weyl)r

(5.2.1)

As in the case of the torus, this space is parameterized by a ﬁnite number of moduli. For the torus, M1 is the upper half-plane mod P S L(2, Z), or equivalently any fundamental region, say F0 . There may also be a subgroup of diﬀ×Weyl, the CKG, that leaves the metric invariant. When there are vertex operators in the path integral it is useful to treat their positions on the same footing as the moduli from the metric, referring to all as moduli. We specify the metric moduli when we need to make the distinction. One way to deal with the CKG, which is applicable if there are vertex operators in the path integral, is to specify the gauge further by ﬁxing the coordinates of some vertex operators. The Polyakov

151

5.2 Moduli and Riemann surfaces

path integral includes an integral over Gr and integrals of the n vertex operator positions over the world-sheet M. The moduli space at topology r with n vertex operators is then Mr,n =

Gr × Mn . (diﬀ × Weyl)r

(5.2.2)

As we have seen for the torus, diﬀr is not in general connected. Picking out the connected component diﬀr0 that contains the identity, the quotient diﬀ r diﬀ r0

(5.2.3)

is the modular group. It is interesting to study the diﬀ×Weyl redundancy of the metric at the inﬁnitesimal level. That is, we are looking for small variations of the metric that are not equivalent to diﬀ×Weyl transformations and so correspond to changes in the moduli. We are also looking for small diﬀ×Weyl transformations that do not change the metric; these would be inﬁnitesimal elements of the CKG, called conformal Killing vectors (CKVs). This will give further insight into the origin of the moduli and the CKG. An inﬁnitesimal diﬀ×Weyl transformation changes the metric by δgab = −2(P1 δσ)ab + (2δω − ∇ · δσ)gab ,

(5.2.4)

where P1 is the traceless symmetric linear combination of derivatives (3.3.17). Moduli correspond to variations δ gab of the metric that are orthogonal to all variations (5.2.4):

0=

=

d2 σ g 1/2 δ gab −2(P1 δσ)ab + (2δω − ∇ · δσ)g ab

d2 σ g 1/2 −2(P1T δ g)a δσ a + δ gab g ab (2δω − ∇ · δσ) . (5.2.5)

The transpose (P1T u)a = −∇b uab is deﬁned as in exercise 3.2. In order for the overlap (5.2.5) to vanish for general δω and δσ we need g ab δ gab = 0 , (P1T δ g)a = 0 .

(5.2.6a) (5.2.6b)

The ﬁrst condition requires that δ gab be traceless, and it is on traceless symmetric tensors that P1T acts. For each solution of these equations there will be a modulus. CKVs are transformations (5.2.4) such that δgab = 0. The trace of this equation ﬁxes δω uniquely, leaving the conformal Killing equation (P1 δσ)ab = 0 .

(5.2.7)

152

5 The string S-matrix

Eqs. (5.2.6) and (5.2.7) become simple for variations around conformal gauge, ∂¯z δ gzz = ∂z δ g¯z¯z = 0, ∂¯z δz = ∂z δ¯z = 0 ,

(5.2.8a) (5.2.8b)

so variations of the moduli correspond to holomorphic quadratic diﬀerentials and CKVs to holomorphic vector ﬁelds. On the torus, the only holomorphic doubly periodic functions are the constants, so there are two real moduli and two real CKVs, in agreement with the discussion in the previous section. The metric moduli correspond to the kernel (5.2.6) of P1T and the CKVs to the kernel (5.2.7) of P1 . The Riemann–Roch theorem relates the number of metric moduli µ = dim ker P1T and the number of CKVs κ = dim ker P1 to the Euler number χ, µ − κ = −3χ .

(5.2.9)

We will be able to derive this later in the chapter. For a closed oriented surface −3χ = 6g − 6. This counting refers to real moduli, separating complex moduli such as τ into their real and imaginary parts. Further, κ vanishes for χ < 0 and µ vanishes for χ > 0. To show this, we ﬁrst cite the fact that it is always possible by a Weyl transformation to ﬁnd a metric for which the scalar curvature R is constant. The sign of R is then the same as that of χ. Now, P1T P1 = − 12 ∇2 − 14 R, so

d2 σ g 1/2 (P1 δσ)ab (P1 δσ)ab =

=

d2 σ g 1/2 δσa (P1T P1 δσ)a

2

d σg

1/2

R 1 ∇a δσb ∇a δσ b − δσa δσ a . 2 4 (5.2.10)

For negative χ the right-hand side is strictly positive, so P1 δσ cannot vanish. A similar argument shows that P1T δ g cannot vanish for positive χ. Combining these results we have χ > 0 : κ = 3χ , µ = 0 , χ < 0 : κ = 0 , µ = −3χ .

(5.2.11a) (5.2.11b)

Riemann surfaces The familiar way to describe moduli space would be to choose one metric from each equivalence class. This would be a family of metrics gˆ ab (t; σ), depending on the moduli tk . For the torus, eq. (5.1.9) gives such a slice. It is often convenient to use an alternative description of the

5.2 Moduli and Riemann surfaces

153

¯ is ﬁxed and the moduli are encoded form (5.1.11), where the metric dwdw in the coordinate region. We can formalize this idea as follows. Let us ﬁrst recall how diﬀerentiable manifolds are deﬁned. One covers the manifold with a set of overlapping patches, with coordinates σma in the mth patch, with a running from 1 to the dimension of the manifold. When patches m and n overlap, the coordinates in the two patches are related by a σma = fmn (σn ) ,

(5.2.12)

where the transition functions fmn are required to be diﬀerentiable a given number of times. For a Riemannian manifold a metric gm,ab (σm ) is also given in each patch, with the usual tensor transformation law relating the values in the overlaps. For a complex manifold, there are complex coordinates zma in each patch, where now a runs from 1 to half the dimension of the manifold. The transition functions are required to be holomorphic, a (zn ) . zma = fmn

(5.2.13)

One can now deﬁne holomorphic functions on the manifold, since the holomorphicity will not depend on which coordinates zma are used. Just as two diﬀerentiable manifolds are equivalent if there is a one-to-one diﬀerentiable map between them, two complex manifolds are equivalent if there is a one-to-one holomorphic map between them. In particular, making a holomorphic change of coordinates within each patch gives an equivalent surface. In the two-dimensional case (one complex coordinate), a complex manifold is known as a Riemann surface. In this case there is a one-to-one correspondence Riemann surfaces ↔ Riemannian manifolds mod Weyl .

(5.2.14)

On the right we put only ‘mod Weyl’ because ‘mod diﬀ’ is already implicit in the deﬁnition of a Riemannian manifold. To see this isomorphism, start with a Riemannian manifold. We know from our discussion of conformal gauge that we can ﬁnd in each coordinate patch a coordinate zm such that ds2 ∝ dzm d¯zm .

(5.2.15)

In neighboring patches the coordinate need not be the same, but because ds2 ∝ dzm d¯zm ∝ dzn d¯zn , the transition functions will be holomorphic.1 This is the map from Riemannian manifolds to complex manifolds in two real dimensions. For the inverse map, one can take the metric dzm d¯zm in 1

To be precise, an antiholomorphic transition function is also possible. For the oriented string we forbid this by choosing a standard orientation; for the unoriented string it is allowed. The term Riemann surface in the strict sense refers to an oriented manifold without boundary.

154

5 The string S-matrix

the mth patch and smooth it in the overlaps (this is a standard construction, the partition of unity) to produce a Riemannian manifold. The two characterizations of a Riemann surface are thus equivalent. The description of the torus in terms of the complex coordinate w of the parallelogram (ﬁgure 5.1) illustrates the idea of a complex manifold. Imagine taking a single coordinate patch which is a little larger than the fundamental parallelogram in ﬁgure 5.1. The periodicity conditions w ∼ = w + 2π ∼ = w + 2πτ

(5.2.16)

are then the transition functions on the overlap between the opposite edges of the patch. The surface is deﬁned by these transition functions. To deﬁne the original path integral over metrics, it is simplest to start by regarding the metric as a function of ﬁxed coordinates such as the square (5.1.7), or at least on a ﬁxed set of coordinate patches with ﬁxed transition functions. To study the quantum ﬁeld theory on a given surface, it then is easiest to work with the unit metric with moduli-dependent transition functions, as in the parallelogram (5.1.11). One can also think of a Riemann surface as follows. Deﬁning the world-sheet as a union of patches, we can use the diﬀ×Weyl freedom to reach the metric dzd¯z in each. The gauge choices on overlapping patches can then diﬀer only by diﬀ×Weyl invariances of this metric, which as we have discussed are the conformal transformations. A Riemann surface is therefore the natural place for a CFT to live, because the ﬁelds in a CFT have deﬁnite transformation properties under conformal transformations. 5.3

The measure for moduli

We now revisit the gauge-ﬁxing of the Polyakov path integral. We have learned that the gauge redundancy does not completely eliminate the path integral over metrics but leaves behind a ﬁnite-dimensional integral over moduli space. It is necessary to reﬁne the discussion from section 3.3 in order to take this into account. The Polyakov path integral for the S-matrix is Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

n [dφ dg] exp(−Sm − λχ) Vdiﬀ×Weyl i=1

d2 σi g(σi )1/2 Vji (ki , σi ) , (5.3.1)

which is the earlier expression (3.5.5) but now written with a general c = ˜c = 26 matter theory, the matter ﬁelds being denoted φ. In gauge-

155

5.3 The measure for moduli

ﬁxing, the integral over metrics and positions is converted to an integral over the gauge group, the moduli, and the unﬁxed positions, [dg] d2n σ → [dζ] dµ t d2n−κ σ .

(5.3.2)

After factoring out the gauge volume, the Jacobian for this transformation becomes the measure on moduli space. The steps are the same as in section 3.3, now taking account of the moduli and the CKVs. In particular, the gauge choice will now ﬁx κ of the vertex operator coordinates, σia → σˆ ia . Label the set of ﬁxed coordinates (a, i) by f. Deﬁne the Faddeev–Popov measure on moduli space by

1 = ∆FP (g, σ)

dµ t diﬀ×Weyl

F

ˆ ζ) [dζ] δ(g − g(t)

δ(σia − σˆ iζa ) .

(5.3.3)

(a,i)∈f

ˆ for at least one By deﬁnition, every metric is (diﬀ×Weyl)-equivalent to g(t) value of t and ζ. The delta function for the ﬁxed coordinates then picks out a unique value of ζ. Actually, as discussed at the end of section 5.1, there may be a residual discrete group of symmetries of ﬁnite order nR , so the delta functions are nonzero at nR points. Inserting the expression (5.3.3) into the path integral (5.3.1) and following the same steps as before leads to Sj1 ...jn (k1 , . . . , kn ) =

compact topologies

F

µ

ˆ σ) ˆ d t ∆FP (g(t),

ˆ × exp −Sm [φ, g(t)] − λχ

n

[dφ]

dσia

(a,i)∈f

ˆ i )1/2 Vji (ki ; σi ) . (5.3.4) g(σ

i=1

The integral over metrics and vertex operator coordinates is now reduced to an integral over the moduli space F for the metric and over the unﬁxed coordinates, with the measure given by ∆FP . In the vertex operators, κ of the positions are ﬁxed. Now we evaluate the Faddeev–Popov measure. The delta functions are nonzero at nR points that are related by symmetry, so we consider one such point and divide by nR . Expand the deﬁnition (5.3.3) of ∆FP near the point. The general metric variation is equal to a local symmetry variation plus a change in the moduli tk , δgab =

µ

ˆ · δσ)gˆ ab . δtk ∂tk gˆ ab − 2(Pˆ 1 δσ)ab + (2δω − ∇

k=1

The inverse Faddeev–Popov determinant is then ˆ σ) ˆ −1 ∆FP (g,

= nR

dµ δt [dδω dδσ] δ(δgab )

δ(δσ a (σˆ i ))

(a,i)∈f

(5.3.5)

156

5 The string S-matrix

= nR

dµ δt dκ x [dβ dδσ]

ˆ + 2πi × exp 2πi(β , 2Pˆ 1 δσ − δtk ∂k g)

a

xai δσ (σˆ i ) ,

(5.3.6)

(a,i)∈f

the inner product in the second line having been deﬁned in exercise 3.2. We have followed the same steps as in the local discussion in section 3.3, writing the delta functions and functionals as integrals over x and βab and be traceless. then integrating out δω to obtain the constraint that βab As before, invert the integral (5.3.6) by replacing all bosonic variables with Grassmann variables: δσ a βab xai δtk

→ → → →

ca , bab , ηai , ξk .

(5.3.7a) (5.3.7b) (5.3.7c) (5.3.7d)

Then with a convenient normalization of the ﬁelds, ˆ σ) ˆ = ∆FP (g,

=

1 nR

1 nR

[db dc] dµ ξ dκ η

1 ˆ + × exp − (b, 2Pˆ 1 c − ξ k ∂k g) ηai ca (σˆ i ) 4π (a,i)∈f

[db dc] exp(−Sg )

µ 1

4π k=1

ˆ (b, ∂k g)

ca (σˆ i ) .

(5.3.8)

(a,i)∈f

In the ﬁnal line we have integrated over the Grassmann parameters ηai and ξ k . The appropriate measure for integration on moduli space is generated by this ghost path integral with insertions. We will not try to keep track of the overall sign at intermediate steps; since this is supposed to be a Jacobian, we implicitly choose the overall sign to give a positive result. The full expression for the S-matrix is then dµ t

Sj1 ...jn (k1 , . . . , kn ) = ×

(a,i)∈f

compact topologies

dσia

F

µ 1

4π k=1

nR

ˆ (b, ∂k g)

[dφ db dc] exp(−Sm − Sg − λχ)

ca (σˆ i )

(a,i)∈f

n

ˆ i )1/2 Vji (ki , σi ) . g(σ

i=1

(5.3.9) This result extends readily to all bosonic string theories — closed and open, oriented and unoriented — the only diﬀerence being which topologies are included and which vertex operators allowed. Eq. (5.3.9) is a useful and

5.3 The measure for moduli

157

elegant result. The complications caused by the moduli and the CKVs are taken into account by the c and b insertions in the path integral. For each ﬁxed coordinate, dσia is replaced by cai , while each metric modulus gives rise to a b insertion. Expression in terms of determinants In eq. (5.3.8) we have represented the Faddeev–Popov determinant as an integral over Grassmann parameters and ﬁelds. We now reduce it to a product of ﬁnite-dimensional and functional determinants. When we go on to study the string S-matrix in coming chapters, this direct path integral approach is just one of the methods we will use. Expand the ghost ﬁelds in suitable complete sets, ca (σ) =

cJ CaJ (σ) ,

bab (σ) =

J

bK BKab (σ) .

(5.3.10)

K

The complete sets CaJ and BKab are deﬁned as follows. The derivative P1 appearing in the ghost action takes vectors into traceless symmetric tensors. Its transpose does the reverse. The ghost action can be written with either, 1 1 T (5.3.11) (b, P1 c) = (P b, c) . 2π 2π 1 We cannot diagonalize P1 in a diﬀ-invariant way because it turns one kind of ﬁeld into another, but we can diagonalize P1T P1 and P1 P1T : Sg =

P1T P1 CaJ = νJ2 CaJ ,

2 P1 P1T BKab = νK BKab .

(5.3.12)

The eigenfunctions can be chosen real, and are normalized in the respective inner products

d2 σ g 1/2 CaJ CJ a = δJJ ,

(5.3.13a)

d2 σ g 1/2 BKab Bab K = δKK .

(5.3.13b)

(P1 P1T )P1 CJ = P1 (P1T P1 )CJ = νJ2 P1 CJ ,

(5.3.14)

(CJ , CJ ) = (BK , BK ) =

Now note that so P1 CJ is an eigenfunction of P1 P1T . In the same way, P1T BK is an eigenfunction of P1T P1 . Thus there is a one-to-one correspondence between the eigenfunctions, except when P1 CJ = 0 or P1T BK = 0. The latter correspond to zero eigenvalues of P1T P1 or P1 P1T . These are just the CKVs and holomorphic quadratic diﬀerentials, and so their numbers are respectively κ and µ. Let us label the eigenfunctions with zero eigenvalue as C0j or B0k , while the nonzero eigenvalues are labeled with J, K = 1, . . ..

158

5 The string S-matrix

For the latter, the normalized eigenfunctions are related BJab =

1 (P1 CJ )ab , νJ

νJ = νJ = 0 .

(5.3.15)

In terms of modes, the ghost path integral ∆FP becomes µ

db0k

κ

dc0j

j=1

k=1

dbJ dcJ exp −

J

µ 1

νJ bJ cJ 2π

4π k =1

ˆ (b, ∂k g)

ca (σi ) .

(a,i)∈f

(5.3.16) From section A.2, a Grassmann integral vanishes unless the variable appears in the integrand. The c0j and b0k do not appear in the action but only in the insertions. In fact, the number of insertions of each type in the integral (5.3.16), κ and µ respectively, just matches the number of ghost zero modes. We have just enough insertions to give a nonzero answer, and only the zero-mode part of the ghost ﬁelds contributes in the insertions. Thus, ∆FP =

µ

db0k

µ b0k

µ k =1

k=1

×

κ

dc0j

j=1

×

J

k =1

ˆ (B0k , ∂k g)

4π

(a,i)∈f

κ

c0j Ca0j (σi )

j =1

νJ bJ cJ dbJ dcJ exp − 2π

.

(5.3.17)

Summing over all ways of saturating the zero-mode integrals with the Grassmann variables generates a ﬁnite-dimensional determinant in each case, while the nonzero modes produce an inﬁnite product, a functional determinant. In all,

∆FP

ˆ (B0k , ∂k g) PTP = det det Ca0j (σi ) det 1 2 1 4π 4π

1/2

.

(5.3.18)

Note that Ca0j (σi ) is a square matrix, in that (a, i) ∈ f runs over κ values as does j. The prime on the functional determinant denotes omission of zero eigenvalues. The Riemann–Roch theorem We can now give a path integral derivation of the Riemann–Roch theorem. For the ghost current of a holomorphic bc system (without ˜b, ˜c), the current conservation anomaly derived in exercise 3.6 is ∇a j a =

1 − 2λ R. 4

(5.3.19)

159

5.4 More about the measure

Noether’s theorem relates conserved currents to invariances. For a current that is not conserved, one ﬁnds by the same argument that δ ([dφ] exp(−S )) i = [dφ] exp(−S ) 2π

d2 σ g 1/2 ∇a j a → −i

2λ − 1 χ. 2

(5.3.20)

The ghost number symmetry acts as δb = −ib, δc = ic. When the path integral is nonvanishing, the transformations of the measure, action, and insertions must cancel. Thus we learn that from the anomaly the number of c minus the number of b insertions is 3χ/2. The path integral calculation relates the number of insertions to the number of zero modes and so gives for the same diﬀerence 1 1 (κ − µ) = (dim ker P1 − dim ker P1T ) . 2 2

(5.3.21)

The factor of 12 appears because we have taken the holomorphic theory only; the antiholomorphic theory gives an equal contribution. Equating the result from the anomaly with the result from the zero modes gives the Riemann–Roch theorem κ − µ = 3χ. For the general bc system one ﬁnds in the same way that dim ker Pn − dim ker PnT = (2n + 1)χ ,

(5.3.22)

the operators being deﬁned in exercise 3.2. 5.4

More about the measure

Here we collect some general properties of the gauge-ﬁxed string amplitude. These are primarily useful for the more formal considerations in chapter 9. Gauge invariance The Faddeev–Popov procedure guarantees that the gauge-ﬁxed amplitude is BRST-invariant and independent of gauge choice, but it is useful to check explicitly that it has all the expected invariance properties. First, it is independent of the choice of coordinates t on moduli space. For new coordinates tk (t), ∂t µ d t = det d t , ∂t µ µ

µ 1 k=1

4π

ˆ = det (b, ∂k g)

∂t ∂t

1 ˆ , (b, ∂k g) 4π k=1

(5.4.1)

160

5 The string S-matrix

and the two Jacobians cancel (up to a sign, which as we have noted must be ﬁxed by hand to give a positive measure). In other words, due to the b-ghost insertions the integrand transforms as a density on moduli space. Second, let us see that the measure is invariant under a Weyl transformation of the gauge slice. Under a Weyl transformation, δ gˆ ab (t; σ) = 2δω(t; σ)gˆ ab (t; σ) ,

(5.4.2)

the variation of the action and measure gives an insertion of the local operator T aa , which vanishes in D = 26 by the equations of motion. We need therefore worry only about the eﬀect on the various insertions. The vertex operator insertions are Weyl-invariant by construction. The insertions ca (σi ) are Weyl-invariant as discussed below eq. (3.3.24). For the b insertions,

(b, ∂k gˆ ) = =

d2 σ gˆ 1/2 bab gˆ ac gˆ bd ∂k gˆ cd

d2 σ gˆ 1/2 bab (gˆ ac gˆ bd ∂k gˆ cd + 2gˆ ab ∂k ω)

ˆ , = (b, ∂k g)

(5.4.3)

the last equality following from tracelessness of b. Third, let us check invariance under an inﬁnitesimal diﬀ transformation, δσ = ξ(t; σ). The extension to a general diﬀ transformation is straightforward. The only terms in the amplitude (5.3.9) that are not immediately invariant are the b insertions and the vertex operators whose coordinates have been ﬁxed. The former transform as ˆ = −2(b, P1 ∂k ξ) δ(b, ∂k g) = −2(P1T b, ∂k ξ) =0

(5.4.4)

using the b equation of motion. The b equation of motion comes from δS /δc = 0, so there will be source terms at the c insertions; these are precisely what is needed to account for the eﬀect of the coordinate transformation on the ﬁxed vertex operators. BRST invariance We now verify the full BRST invariance of the amplitude (5.3.9). The path integral measure and action are known to be invariant from the local analysis, so we have to consider the eﬀect of the BRST transformation on the insertions. In the gauge-ﬁxed path integral (5.3.9), the unintegrated vertex operators are accompanied by a factor of c or ˜cc. This is precisely what we described at the end of section 4.4 for the BRST-invariant vertex operator corresponding to an OCQ state. The integrated vertex operators,

5.4 More about the measure

161

on the other hand, are not BRST-invariant, but we leave it as an exercise to show that they have the BRST variation δB Vm = i∂a (ca Vm ) ,

(5.4.5)

which vanishes upon integration. Finally there is the variation of the b insertion, ˆ = i(T , ∂k g) ˆ . δB (b, ∂k g)

(5.4.6)

This insertion of the energy-momentum tensor just produces a derivative with respect to the modulus tk , so the BRST variation vanishes up to a possible term from the boundary of moduli space. We will examine the boundaries of moduli space in the coming chapters. In most cases there is no surface contribution, due to what is known as the canceled propagator argument. In certain circumstances this argument does not apply, and we will see how to deal with this. It is important to note that the gauge-ﬁxed result (5.3.9) could be written down directly from the requirement of BRST invariance, without reference to the gauge-invariant form. As we saw when evaluating the ghost path integral, there must be at least µ b insertions and κ c insertions, or else the path integral vanishes. In the amplitude (5.3.9) there are precisely the right number of ghost insertions to give a nonzero result. Once we include the necessary b factors, the BRST variation brings in the energymomentum tensor convoluted with the tk -derivative of the metric. This is proportional to a derivative with respect to the moduli, so we get an invariant result only by integrating over moduli space as we have done. The result (5.3.9) can be generalized in various ways, for example to the case that part of the conformal Killing invariance is left unﬁxed. We will illustrate this explicitly in chapter 7 for the example of the torus with no vertex operators. BRST invariance implies that amplitudes for BRST-equivalent states are equal. If we add to any state a null piece QB |χ , the eﬀect is to insert in the path integral the variation δB Vχ ; this integrates to zero. This is important because the physical Hilbert space is identiﬁed with the cohomology, so equivalent states should have equal amplitudes. Measure for Riemann surfaces We derived the Faddeev–Popov measure in the framework where the gauge-ﬁxed world-sheet is described by a moduli-dependent metric. We now recast the result in the Riemann surface framework, where the structure is encoded in moduli-dependent transition functions. To express the measure at a given point t0 in moduli space, let us take a set of coordinate patches, with complex coordinate zm in the mth patch

162

5 The string S-matrix

ˆ 0 ) is Weyland holomorphic transition functions, where the metric g(t equivalent to dzm d¯zm in each patch. Consider now a change in the moduli. We will ﬁrst describe this as a change in the metric with ﬁxed transition functions, and then convert this to a change in the transition functions with ﬁxed metric. In the ﬁrst description, deﬁne the Beltrami diﬀerential 1 µkab = gˆ bc ∂k gˆ ac . 2

(5.4.7)

The b insertion for δtk becomes 1 1 d2 z bzz µk¯zz + b¯z¯z µkz¯z . (b, µk ) = 2π 2π

(5.4.8)

In the second description, after a change δtk in the moduli there will be new coordinates in each patch zm (zm , ¯zm ) . zm = zm + δtk vkm

(5.4.9)

a is a vector index; note that v a is deﬁned only in The superscript on vkm km th the m patch. By the deﬁnition of a Riemann surface, dzm d¯zm is Weylequivalent to the metric at t0 + δt,

dzm d¯zm ∝ dzm d¯zm + δtk µkzm¯zm dzm dzm + µk¯zmzm d¯zm d¯zm

.

(5.4.10)

a is thus related to the Beltrami diﬀerential by The coordinate change vkm ¯zm , µkzm¯zm = ∂zm vkm

zm µk¯zmzm = ∂¯zm vkm .

(5.4.11)

zm is This is the inﬁnitesimal version of Beltrami’s equation. It implies that vkm not holomorphic — otherwise, it would not correspond to a change in the zm has a holomorphic part not determined by Beltrami’s moduli. Also, vkm ¯zm equation, and vkm an antiholomorphic part; these just correspond to the freedom to make holomorphic reparameterizations within each patch. Integrating by parts, the b insertion (5.4.8) becomes

1 1 (b, µk ) = 2π 2πi m

Cm

¯zm zm dzm vkm bzm zm − d¯zm vkm b¯zm ¯zm

,

(5.4.12)

the contour Cm circling the mth patch counterclockwise. By eq. (5.4.9), the derivative of the coordinate with respect to the moduli at a given point is dzm zm = vkm . (5.4.13) dtk The change in the transition functions under a change in moduli is therefore ∂zm ∂zm zn zm zm zm = v − v = vkm − vkn . (5.4.14) km ∂tk zn ∂zn t kn

163

5.4 More about the measure

z

z' 0 C

Fig. 5.3. Coordinate patches z (diagonal hatching) and z (horizontal hatching) with vertex operator at z = 0. In the annular region, z = z + zv .

The contour integrals (5.4.12) around adjacent patches then combine to give 1 1 (b, µk ) = 2π 2πi (mn)

Cmn

∂zm ∂¯zm dzm k bzm zm − d¯zm k b¯zm ¯zm ∂t zn ∂t zn

,

(5.4.15)

which is proportional to the derivatives of the transition functions. The sum runs over all pairs of overlapping patches. The contour Cmn runs between patches m and n, counterclockwise from the point of view of m. The (mn) term is symmetric in m and n, though this is not manifest. Each contour either closes or ends in a triple overlap of patches m, n, and p, where the contours Cmn , Cnp , and Cpm meet at a point. The b insertion is now expressed entirely in terms of the data deﬁning the Riemann surface, the transition functions modulo holomorphic equivalence. As an illustration, we can use this to put the moduli for the metric and for the vertex operator positions on a more equal footing. Consider a vertex operator at position zv in a coordinate system z. Let us make a small new coordinate patch z centered on the vertex operator, as shown in ﬁgure 5.3. We keep the vertex operator at coordinate point z = 0, and encode its position in the transition functions z = z + zv

(5.4.16)

for the annular overlap region. The measure for the moduli zv , ¯zv is given

164

5 The string S-matrix

by the form (5.4.15), where

∂z = −1 . ∂zv z

(5.4.17)

Thus the ghost insertion is C

dz bz z 2πi

C

d¯zm b¯z ¯z = b−1 ˜b−1 · , −2πi

(5.4.18)

where C is any contour encircling the vertex operator as shown in the ﬁgure. The full expression for the S-matrix is then compactly written as S (1; . . . ; n) =

compact topologies

−λχ

e

F

dm t nR

m

Bk

k=1

n

ˆi V

.

(5.4.19)

i=1

ˆ Here Bk is the abbreviation for the b ghost insertion (5.4.15), and V a denotes ˜ccVm for a closed string or ta c Vm for an open string. That is, we are now treating all vertex operators as ﬁxed, having traded their coordinates for extra parameters in the transition functions. The number m of moduli is m = µ + 2nc + no − κ = −3χ + 2nc + no ,

(5.4.20)

where nc and no are the number of closed and open string vertex operators respectively. The expression (5.4.19) shows that the hatted vertex operator, with ˜cc or ta ca , is the basic one. This is what comes out of the state–operator correspondence, and it is BRST-invariant. If the vertex operator is integrated, the ghost insertions (5.4.18) simply remove the ˜cc or ta ca . For example, b−1 ˜b−1 · ˜ccVm = Vm ,

(5.4.21)

leaving the integrated form of the vertex operator. We derived the result (5.4.19) from the Polyakov path integral, so the vertex operators come out in OCQ form, but we may now use arbitrary BRST-invariant vertex operators. Exercises 5.1 Consider the sum (5.1.1), but now over all particle paths beginning and ending at given points in spacetime. In this case ﬁxing the tetrad leaves no coordinate freedom. (a) Derive the analog of the gauge-ﬁxed path integral (5.3.9). (b) Reduce the ghost path integral to determinants as in eq. (5.3.18). (c) Reduce the X µ path integral to determinants (as will be done in section 6.2 for the string).

Exercises

165

(d) Evaluate the ﬁnite and functional determinants and show that the result is the scalar propagator for a particle of mass m. 5.2 Repeat the steps in the previous problem for paths forming a closed loop. In order to ﬁx the residual gauge symmetry, introduce a fake vertex operator by diﬀerentiating with respect to m2 . The answer is given as eq. (7.3.9). If you do both exercises 5.1 and 5.2, discuss the relation between the results for the moduli measure. 5.3 (a) Show from the OPE with the BRST current that the vertex operator ˜ccVm is BRST-invariant for Vm a (1,1) matter tensor. (b) Show that the vertex operator satisﬁes eq. (5.4.5).

6 Tree-level amplitudes

We are now ready to study string interactions. In this chapter we consider the lowest order amplitudes, coming from surfaces with positive Euler number. We ﬁrst describe the relevant Riemann surfaces and calculate the CFT expectation values that will be needed. We next study scattering amplitudes, ﬁrst for open strings and then for closed. Along the way we introduce an important generalization in the open string theory, the Chan–Paton factors. At the end of the section we return to CFT, and discuss some general properties of expectation values. 6.1

Riemann surfaces

There are three Riemann surfaces of positive Euler number: the sphere, the disk, and the projective plane. The sphere The sphere S2 can be covered by two coordinate patches as shown in ﬁgure 6.1. Take the disks |z| < ρ and |u| < ρ for ρ > 1 and join them by identifying points such that u = 1/z .

(6.1.1)

In fact, we may as well take ρ → ∞. The coordinate z is then good everywhere except at the ‘north pole,’ u = 0. We can work mainly in the z patch except to check that things are well behaved at the north pole. We can think of the sphere as a Riemann surface, taking the ﬂat metric in both patches and connecting them by a conformal (coordinate plus Weyl) transformation. Or, we can think of it as a Riemannian manifold, with a globally deﬁned metric. A general conformal gauge metric is ds2 = exp(2ω(z, ¯z ))dzd¯z . 166

(6.1.2)

6.1 Riemann surfaces

167

0 u

z 0 Fig. 6.1. The sphere built from z and u coordinate patches.

Since dzd¯z = |z|4 dud¯u, the condition for the metric to be nonsingular at u = 0 is that exp(2ω(z, ¯z )) fall as |z|−4 for z → ∞. For example, ds2 =

4r2 dzd¯z 4r2 dud¯u = (1 + z¯z )2 (1 + u¯u)2

(6.1.3)

describes a sphere of radius r and curvature R = 2/r2 . According to the general discussion in section 5.2, the sphere has no moduli and six CKVs, so that in particular every metric is (diﬀ×Weyl)equivalent to the round metric (6.1.3). Let us see this at the inﬁnitesimal level. As in eq. (5.2.8) one is looking for holomorphic tensor ﬁelds δgzz (z) and holomorphic vector ﬁelds δz(z). These must be deﬁned on the whole sphere, so we need to consider the transformation to the u-patch, ∂u (6.1.4a) δz = −z −2 δz , ∂z −2 ∂u δguu = δgzz = z 4 δgzz . (6.1.4b) ∂z Any holomorphic quadratic diﬀerential δgzz would have to be holomorphic in z but vanish as z −4 at inﬁnity, and so must vanish identically. A CKV δz, on the other hand, is holomorphic at u = 0 provided it grows no more rapidly than z 2 as z → ∞. The general CKV is then δu =

δz = a0 + a1 z + a2 z 2 , δ¯z = a∗0 + a∗1 ¯z + a∗2 ¯z 2 ,

(6.1.5a) (6.1.5b)

with three complex or six real parameters as expected from the Riemann– Roch theorem. These inﬁnitesimal transformations exponentiate to give the M¨obius

168

6 Tree-level amplitudes

group z =

αz + β γz + δ

(6.1.6)

for complex α, β, γ, δ. Rescaling α, β, γ, δ leaves the transformation unchanged, so we can ﬁx αδ − βγ = 1 and identify under an overall sign reversal of α, β, γ, δ. This deﬁnes the group PSL(2, C). This is the most general coordinate transformation that is holomorphic on all of S2 . It is one-to-one with the point at inﬁnity included. Three of the six parameters correspond to ordinary rotations, forming an S O(3) subgroup of PSL(2, C). The disk It is useful to construct the disk D2 from the sphere by identifying points under a reﬂection. For example, identify points z and z such that z = 1/¯z .

(6.1.7)

In polar coordinates z = reiφ , this inverts the radius and leaves the angle ﬁxed, so the unit disk |z| ≤ 1 is a fundamental region for the identiﬁcation. The points on the unit circle are ﬁxed by the reﬂection, so this becomes a boundary. It is often more convenient to use the conformally equivalent reﬂection z = ¯z .

(6.1.8)

The upper half-plane is now a fundamental region, and the real axis is the boundary. The CKG of the disk is the subgroup of P S L(2, C) that leaves the boundary of the disk ﬁxed. For the reﬂection (6.1.8) this is just the subgroup of (6.1.6) with α, β, γ, δ all real, which is P S L(2, R), the M¨ obius group with real parameters. One CKV is the ordinary rotational symmetry of the disk. Again, all metrics are equivalent — there are no moduli. The projective plane The projective plane RP2 can also be obtained as a Z2 identiﬁcation of the sphere. Identify points z and z with z = −1/¯z .

(6.1.9)

These points are diametrically opposite in the round metric (6.1.3). There are no ﬁxed points and so no boundary in the resulting space, but the space is not oriented. One fundamental region for the identiﬁcation is the unit disk |z| ≤ 1, with points eiφ and −eiφ identiﬁed. Another choice is the upper half-z-plane. There are no moduli. The CKG is the subgroup of

169

6.2 Scalar expectation values

P S L(2, C) that respects the identiﬁcation (6.1.9); this is just the ordinary rotation group S O(3). Both the disk and projective plane have been represented as the sphere with points identiﬁed under a Z2 transformation, or involution. In fact, every world-sheet can be obtained from a closed oriented world-sheet by identifying under one or two Z2 s. The method of images can then be used to obtain the Green’s functions. 6.2

Scalar expectation values

The basic quantities that we need are the expectation values of products of vertex operators. In order to develop a number of useful techniques and points of view we will calculate these in three diﬀerent ways: by direct path integral evaluation, by using holomorphicity properties, and later in the chapter by operator methods. The path integral method has already been used in section 5.3 for the Faddeev–Popov determinant and in the appendix for the harmonic oscillator. Start with a generating functional -

Z[J] =

exp i

.

2

(6.2.1)

d σ J(σ)·X(σ)

for arbitrary Jµ (σ). For now we work on an arbitrary compact twodimensional surface M, and in an arbitrary spacetime dimension d. Expand X µ (σ) in terms of a complete set XI (σ), X µ (σ) =

µ

xI XI (σ) ,

(6.2.2a)

I

∇2 XI = −ωI2 XI ,

M

(6.2.2b)

d2 σg 1/2 XI XI = δII .

(6.2.2c)

Then Z[J] =

dxµI

I,µ

where JIµ =

ωI2 xIµ xIµ exp − + ixµI JIµ 4πα

,

d2 σJ µ (σ)XI (σ) .

(6.2.3)

(6.2.4)

The integrals are Gaussian except for the constant mode

X0 =

2

d σg

1/2

−1/2

,

(6.2.5)

170

6 Tree-level amplitudes

which has vanishing action and so gives a delta function. Carrying out the integrations leaves d d

Z[J] = i(2π) δ (J0 )

I=0

4π 2 α ωI2

d/2

πα JI · JI exp − ωI2

−d/2

−∇2 = i(2π) δ (J0 ) det 4π 2 α 1 × exp − d2 σ d2 σ J(σ) · J(σ )G (σ, σ ) . 2

d d

(6.2.6)

As discussed in section 2.1 and further in section 3.2, the timelike modes x0I give rise to wrong-sign Gaussians and are deﬁned by the contour rotation1 x0I → −ixdI , I = 0. The primed Green’s function excludes the zero mode contribution, G (σ1 , σ2 ) =

2πα I=0

ωI2

XI (σ1 )XI (σ2 ) .

(6.2.7)

It satisﬁes the diﬀerential equation −

1 2 ∇ G (σ1 , σ2 ) = XI (σ1 )XI (σ2 ) 2πα I=0

= g −1/2 δ 2 (σ1 − σ2 ) − X20 ,

(6.2.8)

where the completeness of the XI has been used. The ordinary Green’s function with a delta function source does not exist. It would correspond to the electrostatic potential of a single charge, but on a compact surface the ﬁeld lines from the source have no place to go. The X20 term can be thought of as a neutralizing background charge distribution. The sphere Specializing to the sphere, the solution to the diﬀerential equation (6.2.8) is G (σ1 , σ2 ) = −

1

α ln |z12 |2 + f(z1 , ¯z1 ) + f(z2 , ¯z2 ) , 2

(6.2.9)

A curious factor of i has been inserted into eq. (6.2.6) by hand because it is needed in the S-matrix, but it can be understood formally as arising from the same rotation. If we rotated the entire ﬁeld X 0 → −iX d , the Jacobian should be 1, by the usual argument about rescaling ﬁelds (footnote 1 of the appendix). However, we do not want to rotate x00 , because this mode produces the energy delta function. So we have to rotate it back, giving a factor of i.

171

6.2 Scalar expectation values where

α X20 d2 z exp[2ω(z , ¯z )] ln |z − z |2 + k . (6.2.10) 4 The constant k is determined by the property that G is orthogonal to X0 , but in any case we will see that the function f drops out of all expectation values. It comes from the background charge, but the delta function from the zero-mode integration forces overall neutrality, J0µ = 0, and the background makes no net contribution. Now consider the path integral with a product of tachyon vertex operators, f(z, ¯z ) =

AnS2 (k, σ) =

eik1 ·X(σ1 )

r

eik2 ·X(σ2 ) . . . eikn ·X(σn )

r

r

S2

(6.2.11)

.

This corresponds to J(σ) =

n

ki δ 2 (σ − σi ) .

(6.2.12)

i=1

The amplitude (6.2.6) then becomes AnS2 (k, σ) = iCSX2 (2π)d δ d (

i ki )

n

n 1 ki ·kj G (σi , σj ) − ki2 Gr (σi , σi ) × exp − 2 i,j=1 i=1

(6.2.13)

.

i<j

The constant here is

CSX2 = X−d 0

−∇2 det 2 4π α

−d/2

(6.2.14)

. S2

The determinant can be regularized and computed, but we will not need to do this explicitly. We are employing here the simple renormalization used in section 3.6, so that the self-contractions involve Gr (σ, σ ) = G (σ, σ ) +

α ln d2 (σ, σ ) . 2

(6.2.15)

Note that Gr (σ, σ) = 2f(z, ¯z ) + α ω(z, ¯z )

(6.2.16)

is ﬁnite by design. The path integral on the sphere is then

α 2 AnS2 (k, σ) = iCSX2 (2π)d δ d ( i ki ) exp − k ω(σi ) 2 i i

n

|zij |α ki ·kj .

i,j=1 i<j

(6.2.17) The function f has dropped out as promised. The dependence on the conformal factor ω(σ) is precisely that found in section 3.6 from the Weyl

172

6 Tree-level amplitudes

anomaly in the vertex operator. It will cancel the variation of g 1/2 for an on-shell operator.2 We can take a metric that is ﬂat in a large region containing all the vertex operators (‘pushing the curvature to inﬁnity’) and the term in ω(σi ) drops out. Higher vertex operators are exponentials times derivatives of X µ , so we also need n

eiki ·X(zi ,¯zi )

i=1

p r

∂X µj (zj )

j=1

q

¯ νk (¯z ) ∂X k

k=1

(6.2.18)

. S2

¯ This is given by summing over all contractions, where every ∂X or ∂X ¯ must be contracted either with an exponential or with another ∂X or ∂X. The XX contraction is simply − 12 α ln |z|2 , the fs again dropping out in the ﬁnal expression. The result can be summarized as iCSX2 (2π)d δ d ( ×

i ki )

p

n

|zij |α ki ·kj

i,j=1 i<j

v

µj

(zj )

+q

µj

j=1

(zj )

q

νk

˜v

(¯zk )

+ q˜

νk

(¯zk )

(6.2.19)

k=1

Here v µ (z) = −i

n α kiµ , 2 i=1 z − zi

˜v µ (¯z ) = −i

n α kiµ 2 i=1 ¯z − ¯zi

(6.2.20)

come from the contractions with the exponentials. The expectation values of q µ = ∂X µ − v µ are given by the sum over all contractions using −η µν (z − z )−2 α /2, and those of q˜ν by the conjugate. Now we repeat the calculation in a diﬀerent way, using holomorphicity. As an example consider

∂X µ (z1 )∂X ν (z2 ) S2 . The OPE determines this to be α η µν

1 S2 + g(z1 , z2 ) , − 2 2z12

(6.2.21)

(6.2.22)

where g(z1 , z2 ) is holomorphic in both variables. In the u-patch, ∂u X µ = −z 2 ∂z X µ ,

(6.2.23)

so the condition of holomorphicity at u = 0 is that expectation values of ∂z X µ fall as z −2 at inﬁnity. More generally, a tensor of weight (h, 0) must 2

The constant CSX also depends on the conformal factor, because the X µ CFT by itself has 2 nonzero central charge, but in the full string theory this will cancel.

173

6.2 Scalar expectation values

behave as z −2h at inﬁnity. Focusing on the z1 -dependence in eq. (6.2.22) at ﬁxed z2 , this implies that g(z1 , z2 ) falls as z1−2 at inﬁnity and so vanishes by holomorphicity. This determines the expectation value up to normalization. Comparing with the path integral result (6.2.19), there is agreement, though the normalization 1 S2 is seen to diverge as δ(0). This is just the zero-mode divergence from the inﬁnite volume of spacetime. To obtain the expectation value of the product of vertex operators AnS2 (k, σ) =

n

: eiki ·X(zi ,¯zi ) :

i=1

(6.2.24) S2

by this method is less direct, because the exponentials are not holomorphic. In fact they can be factored into holomorphic and antiholomorphic parts, but this is subtle and is best introduced in operator language as we will do in chapter 8. We use the holomorphicity of the translation current. Consider the expectation value with one current ∂X µ (z) added. The OPE of the current with the vertex operators determines the singularities in z, µ

∂X (z)

n

iki ·X(zi ,¯zi )

:e

i=1

=−

: S2

n iα n kiµ AS2 (k, σ) 2 z − zi i=1

+ terms holomorphic in z.

(6.2.25)

Now look at z → ∞. The condition that ∂u X µ be holomorphic at u = 0 again requires that the expectation value vanish as z −2 when z → ∞. The holomorphic term in eq. (6.2.25) must then vanish, and from the vanishing of the order z −1 term we recover momentum conservation, AnS2 (k, σ)

n µ

ki = 0.

(6.2.26)

i=1

Let us also say this in a slightly diﬀerent way. Consider the contour integral of the spacetime translation current 1 p = 2πi

µ

C

(dz jzµ − d¯z j¯zµ ) ,

(6.2.27)

where the contour C encircles all the exponential operators. There are two ways to evaluate this. The ﬁrst is to contract C until it is a small circle around each vertex operator, which picks out the (z − zi )−1 term from each OPE and gives ni=1 kiµ . The second is to expand it until it is a small circle in the u-patch: by holomorphicity at u = 0 it must vanish. This sort of ‘contour-pulling’ argument is commonly used in CFT.

174

6 Tree-level amplitudes

We have used the OPE to determine the singularities and then used holomorphicity to get the full z-dependence. Now we look at the second term of the OPE as z → z1 . Expanding the result (6.2.25) gives

n iα k1µ kiµ − AnS2 (k, σ) + + O(z − z1 ) . 2 z − z1 i=2 z1 − zi

(6.2.28)

Contracting this with ik1µ , the term of order (z − z1 )0 must agree with the OPE ik1 ·∂X(z) : eik1 ·X(z1 ,¯z1 ) : α k12 = : eik1 ·X(z1 ,¯z1 ) : + ∂z1 : eik1 ·X(z1 ,¯z1 ) : + O(z − z1 ) . 2(z − z1 )

(6.2.29)

This implies ∂z1 AnS2 (k, σ) =

n k1 ·ki α n . AS2 (k, σ) 2 z1i i=2

(6.2.30)

Integrating, and using the conjugate equation and momentum conservation, determines the path integral up to normalization, AnS2 (k, σ) ∝ δ d (

i ki )

n

|zij |α ki ·kj ,

(6.2.31)

i,j=1 i<j

in agreement with the ﬁrst method. Note that to compare we must push the curvature to inﬁnity (ω(σi ) = 0) so that [ ]r = : :. It is worth noting that the intermediate steps in the path integral method depend in a detailed way on the particular choice of Riemannian metric. The metric is needed in order to preserve coordinate invariance in these steps. A diﬀerent, Weyl-equivalent, metric gives a diﬀerent Laplacian and diﬀerent eigenfunctions, though at the very end this dependence must drop out in string theory. The second method uses only the basic data of a Riemann surface, its holomorphic structure. The disk The generalization to the disk is straightforward. We will represent the disk by taking the above representation of the sphere and restricting z to the upper half of the complex plane. The Neumann boundary term is accounted for by an image charge, G (σ1 , σ2 ) = −

α α ln |z1 − z2 |2 − ln |z1 − ¯z2 |2 , 2 2

(6.2.32)

175

6.2 Scalar expectation values up to terms that drop out due to momentum conservation. Then n

iki ·X(zi ,¯zi )

:e

= iCDX2 (2π)d δ d (

:

i=1

i ki )

n

2

|zi − ¯zi |α ki /2

i=1

D2

×

n

|zi − zj |α ki ·kj |zi − ¯zj |α ki ·kj .

(6.2.33)

i,j=1 i<j

For expectation values with ∂a X µ s one again sums over contractions, ¯ ν (¯z ) (or now using the Green’s function (6.2.32). Note that ∂X µ (z)∂X µ ν q (¯z )) has a nonzero contraction. q (z)˜ For two points on the boundary, the two terms in the Green’s function (6.2.32) are equal, and the Green’s function diverges at zero separation even after normal ordering subtracts the ﬁrst term. For this reason, boundary operators must be deﬁned with boundary normal ordering, where the subtraction is doubled:

X µ (y1 )X ν (y2 ) = X µ (y1 )X ν (y2 ) + 2α η µν ln |y1 − y2 | ,

(6.2.34)

with y denoting a coordinate on the real axis. The combinatorics are the same as for other forms of normal ordering. Boundary normal-ordered expectation values of boundary operators have the same good properties (nonsingularity) as conformal normal-ordered operators in the interior. An expectation value with exponentials both in the interior and on the boundary, each having the appropriate normal ordering, is given by taking the appropriate limit of the interior result (6.2.33) and dropping the 2 |zi − ¯zi |α ki /2 factor for the boundary operators. Explicitly, for exponentials all on the boundary, n

iki ·X(yi )

e

= iCDX2 (2π)d δ d (

i=1

i ki )

More generally,

eiki ·X(yi )

i=1

p

(6.2.35)

∂y X µj (yj )

j=1

×

|yi − yj |2α ki ·kj .

i,j=1 i<j

D2

n

n

n i,j=1 i<j

|yij |

2α ki ·kj

= iCDX2 (2π)d δ d (

D p2

v

µj

(yj )

+q

j=1

µj

(yj )

i ki )

,

(6.2.36)

D2

where now v µ (y) = −2iα

n i=1

kiµ y − yi

and the qs are contracted using −2α (y − y )−2 η µν .

(6.2.37)

176

6 Tree-level amplitudes The projective plane

The method of images gives the Green’s function as G (σ1 , σ2 ) = − Now

n

α α ln |z1 − z2 |2 − ln |1 + z1 ¯z2 |2 . 2 2

iki ·X(zi ,¯zi )

:e

:

i=1

=

X iCRP (2π)d δ d ( 2

i ki )

n

(6.2.38)

2

|1 + zi ¯zi |α ki /2

i=1

RP2

×

n

|zi − zj |α ki ·kj |1 + zi ¯zj |α ki ·kj ,

(6.2.39)

i,j=1 i<j

and so on for more general expectation values. There is no boundary because the involution has no ﬁxed points — there is no way for a point z to approach its image −¯z −1 . 6.3

The bc CFT The sphere

The path integral for the ghosts has already been set up in section 5.3. According to the Riemann–Roch theorem, the simplest nonvanishing expectation value will be

c(z1 )c(z2 )c(z3 )˜c(¯z4 )˜c(¯z5 )˜c(¯z6 ) S2 .

(6.3.1)

Up to normalization (a functional determinant), the result (5.3.18) was just the zero-mode determinant, det Ca0j (σi ) .

(6.3.2)

The six CKVs were found in section 6.1. In a complex basis they are Cz = 1, z, z 2 , C¯z = 0 , Cz = 0 , C¯z = 1, ¯z , ¯z 2 .

(6.3.3a) (6.3.3b)

In this basis the determinant splits into two 3 × 3 blocks, and the expectation value becomes 1 CSg2 det z1 z2 1

1 1 1 1 1 z2 z3 det ¯z4 ¯z5 ¯z6 = CSg2 z12 z13 z23 ¯z45 ¯z46 ¯z56 . (6.3.4) ¯ z42 ¯z52 ¯z62 z22 z32

The constant CSg2 includes the functional determinant and also a ﬁnitedimensional Jacobian (independent of the positions) that arises because the basis (6.3.3) is not orthonormal.

177

6.3 The bc CFT

This is the only bc path integral that we will actually need for the tree-level amplitudes, but for completeness we give p+3 i=1

c(zi )

p

b(zj )

= CSg2

· (anti)

j=1

S2

zp+1,p+2 zp+1,p+3 zp+2,p+3 · (anti) (z1 − z1 ) . . . (zp − zp ) ± permutations ,

(6.3.5)

obtained by contracting the bs with cs. The antiholomorphic part ‘anti’ has the same form. We are being a little careless with the overall sign; again, for the Faddeev–Popov determinant we will in any case take an absolute value. To give an alternative derivation using holomorphicity, we again consider ﬁrst the conservation law, derived /by inserting into the amplitude (6.3.5) the ghost number contour integral C dz j(z)/2πi, where C encircles all the vertex operators and j = − : bc : is again the ghost number current. From the OPE with b and c, this just counts the number of c minus the number of b ﬁelds, giving nc − nb times the original amplitude. Now pull the contour into the u-patch using the conformal transformation (2.5.17), C

dz jz = − 2πi

C

du ju + 3 → 3 . 2πi

(6.3.6)

The shift +3 comes about because j is not a tensor, having a z −3 term in its OPE with T . In the last step we use holomorphicity in the u-patch. The nonzero amplitudes therefore have nc − nb = 3 and similarly n˜c − n˜b = 3, in agreement with the Riemann–Roch theorem. From the OPE, the ghost expectation value (6.3.1) is holomorphic in each variable, and it must have a zero when two identical anticommuting ﬁelds come together. Thus it must be of the form ˜ z4 , ¯z5 , ¯z6 ) , z12 z13 z23 ¯z45 ¯z46 ¯z56 F(z1 , z2 , z3 )F(¯

(6.3.7)

˜ antiholomorphic functions of the positions. with F holomorphic and F As z1 → ∞ this goes as z12 F. However, c is a tensor of weight −1, so the amplitude can be no larger than z12 at inﬁnity. Thus F(zi ) must be ˜ we independent of z1 , and so also of z2 and z3 ; arguing similarly for F obtain again the result (6.3.4). The same argument gives for the general case (6.3.5) the result CSg2

p+3 i,i =1 i

zii

p j,j =1 j<j

zjj

p p+3

(zi − zj )−1 · (anti) ,

(6.3.8)

i=1 j=1

which has the correct poles, zeros, and behavior at inﬁnity. The permutations (6.3.5) evidently sum up to give this single term.

178

6 Tree-level amplitudes

We have considered the λ=2 theory that is relevant to the ghosts of the bosonic string, but either method generalizes readily to any λ. The disk The simplest way to obtain the bc amplitudes on the disk is with the doubling trick. As in eq. (2.7.30), we can represent both the holomorphic and antiholomorphic ﬁelds in the upper half-plane by holomorphic ﬁelds in the whole plane, using ˜b(¯z ) = b(z ) , ˜c(¯z ) = c(z ) , z = ¯z , Im z > 0 . (6.3.9) The expectation value of the holomorphic ﬁelds follows as on the sphere,

c(z1 )c(z2 )c(z3 ) D2 = CDg 2 z12 z13 z23 for all z. Then for example

c(z1 )c(z2 )˜c(¯z3 ) D2 = c(z1 )c(z2 )c(z3 )

(6.3.10)

D2

= CDg 2 z12 (z1 − ¯z3 )(z2 − ¯z3 ) .

(6.3.11)

More general correlators are obtained in the same way. The projective plane The doubling trick again can be used. The involution z = −¯z −1 implies that ˜b(¯z ) =

∂z ∂¯z

2

b(z ) = z 4 b(z ) ,

˜c(¯z ) =

∂z ∂¯z

−1

c(z ) = z −2 c(z ) . (6.3.12)

Again g

c(z1 )c(z2 )c(z3 ) RP2 = CRP z z z 2 12 13 23

and so

c(z1 )c(z2 )˜c(¯z3 ) RP2 = z3−2 c(z1 )c(z2 )c(z3 ) = 6.4

g CRP z (1 2 12

(6.3.13)

RP2

+ z1 ¯z3 )(1 + z2 ¯z3 ) .

(6.3.14)

The Veneziano amplitude

Open string amplitudes are slightly simpler than closed string amplitudes, so we begin with these. We represent the disk as the upper half-plane so the boundary coordinate y is real. There are no moduli. After ﬁxing the metric, the CKG P S L(2, R) can be used to ﬁx the three vertex operators to arbitrary positions y1 , y2 , y3 on the boundary, except that this group does not change

179

6.4 The Veneziano amplitude

the cyclic ordering of the vertex operators so we must sum over the two orderings. For three open string tachyons on the disk, the general expression (5.3.9) for the string S-matrix thus reduces to SD2 (k1 ; k2 ; k3 ) = go3 e−λ

c1 eik1 ·X (y1 ) c1 eik2 ·X (y2 ) c1 eik3 ·X (y3 )

+ (k2 ↔ k3 ) ,

D2

(6.4.1)

each ﬁxed coordinate integration being replaced by the corresponding c-ghost. Each vertex operator includes a factor of go , the open string coupling. The factor e−λ is from the Euler number term in the action. Of course go and eλ are related, go2 ∝ eλ , but we will determine the constant of proportionality as we go along. The expectation values in (6.4.1) were found in the previous sections (again, we take the absolute value for the ghosts), giving SD2 (k1 ; k2 ; k3 ) = igo3 CD2 (2π)26δ 26 ( ×|y12 |

1+2α k1 ·k2

|y13 |

i ki ) 1+2α k1 ·k3

|y23 |1+2α k2 ·k3 + (k2 ↔ k3 ), (6.4.2)

where CD2 = e−λ CDX2 CDg 2 . Momentum conservation and the mass-shell condition ki2 = 1/α imply that 2α k1 ·k2 = α (k32 − k12 − k22 ) = −1

(6.4.3)

and the same for the other ki · kj , so this reduces to SD2 (k1 ; k2 ; k3 ) = 2igo3 CD2 (2π)26δ 26 (

i ki )

(6.4.4)

.

This is independent of the gauge choice yi , which is of course a general property of the Faddeev–Popov procedure. The Weyl invariance is crucial here — if the vertex operators were taken oﬀ the mass shell, the amplitude would depend on the choice of yi . We could have used independence of the yi to determine the Faddeev– Popov determinant without calculation. Using the mass-shell condition and momentum conservation, the X µ expectation value is proportional to |y12 y13 y23 |−1 , and so the measure must be reciprocal to this. This same measure then applies for n > 3 vertex operators, because in all cases three positions are ﬁxed. The four-tachyon amplitude is obtained in the same way, SD2 (k1 ; k2 ; k3 ; k4 ) = go4 e−λ =

∞

−∞

dy4

3

igo4 CD2 (2π)26δ 26 (

c1 (yi )eiki ·X(yi ) eik4 ·X(y4 )

i=1 i ki )|y12 y13 y23 |

∞ −∞

dy4

+ (k2 ↔ k3 )

|yij |2α ki ·kj

i<j

+ (k2 ↔ k3 ) .

(6.4.5)

180

6 Tree-level amplitudes

3

2

3

2

3

4

(a) 4

1

(b) 1

4

(c) 1

2

2

3

2

3

2

4

(d) 4

1

(e) 1

4

(f) 1

3

Fig. 6.2. The six cyclically inequivalent orderings of four open string vertex operators on the disk. The coordinate y increases in the direction of the arrow, except that at point 3 it jumps from +∞ to −∞.

This is again independent of y1,2,3 after a change of variables (a M¨ obius transformation) on y4 . It is convenient to take y1 = 0, y2 = 1, and y3 → ∞. The amplitude is conventionally written in terms of the Mandelstam variables s = −(k1 + k2 )2 ,

t = −(k1 + k3 )2 ,

u = −(k1 + k4 )2 .

(6.4.6)

These are not independent: momentum conservation and the mass-shell condition imply that 4 m2i = − . (6.4.7) s+t+u= α i Using 2α ki · kj = −2 + α (ki + kj )2 , the amplitude becomes SD2 (k1 ; k2 ; k3 ; k4 ) = igo4 CD2 (2π)26δ 26 ( ∞

×

−∞

dy4 |y4 |

−α u−2

i ki )

|1 − y4 |−α t−2 + (t → s) .

(6.4.8)

The integral splits into three ranges, −∞ < y4 < 0, 0 < y4 < 1, and 1 < y4 < ∞. For these three ranges the vertex operators are ordered as in ﬁgures 6.2(a), (b), and (c) respectively. M¨ obius invariance can be used to take each of these ranges into any other, so they give contributions that are equal up to interchange of vertex operators. The (t ↔ s) term gives ﬁgures 6.2(d), (e), and (f). In all, SD2 (k1 ; k2 ; k3 ; k4 ) = 2igo4 CD2 (2π)26δ 26 (

i ki )

I(s, t) + I(t, u) + I(u, s) , (6.4.9)

181

6.4 The Veneziano amplitude

(a)

3

4

2

1

(b)

3

4

1

2

3

4

1

2

(c)

Fig. 6.3. Processes giving poles in the (a) s-, (b) t-, and (c) u-channels.

where

1

I(s, t) = 0

dy y −α s−2 (1 − y)−α t−2 .

(6.4.10)

The three terms come from ﬁgures 6.2(c), (f), 6.2(b), (d), and 6.2(a), (e) respectively. The integral I(s, t) converges if α s < −1 and α t < −1. As α s → −1, the integral diverges at y = 0. To study the divergence, take a neighborhood of y = 0 and approximate the integrand: r

I(s, t) = 0

dy y −α s−2 + terms analytic at α s = −1

r−α s−1 =− + terms analytic at α s = −1 αs+1 1 =− (6.4.11) + terms analytic at α s = −1 . αs+1 In (6.4.11), we have evaluated the integral in the convergent region. We see that the divergence is a pole at s = −1/α , the mass-squared of the open string tachyon. The variable s is just the square of the center-ofmass energy for scattering 1 + 2 → 3 + 4, so this pole is a resonance due to an intermediate tachyon state. Again, it is an artifact of the bosonic string that this lightest string state is tachyonic, and not relevant to the discussion. The pole is due to the process shown in ﬁgure 6.3(a), in which tachyons 1 and 2 join to become a single tachyon, which then splits into tachyons 3 and 4. Because the singularity at α s = −1 is just a pole, I(s, t) can be analytically continued past this point into the region α s > −1. The amplitude is deﬁned via this analytic continuation. The divergence of the amplitude at the pole is an essential physical feature of the amplitude, a resonance corresponding to propagation of the intermediate string state over long spacetime distances. The divergence of the integral past the pole is not; it is just an artifact of this particular integral representation of the amplitude. The continuation poses no problem. In fact, we will see that every

182

6 Tree-level amplitudes

string divergence is of this same basic form, so this one kind of analytic continuation removes all divergences — except of course for the poles themselves. The pole is on the real axis, so we need to deﬁne it more precisely. The correct prescription for a Minkowski process is 1 1 ≡ +1 α s + 1 + i 1 − iπδ(α s + 1) , (6.4.12) ≡P αs+1 where P denotes the principal value. Unitarity (which we will develop more systematically in chapter 9) requires this pole to be present and determines its coeﬃcient in terms of the amplitude for two tachyons to scatter into one: d26 k SD2 (k1 ; k2 ; k)SD2 (−k; k3 ; k4 ) SD2 (k1 ; k2 ; k3 ; k4 ) = i (2π)26 −k 2 + α−1 + i α s

+ terms analytic at k 2 = 1/α . (6.4.13) Gathering together the factors in the four-tachyon amplitude, including an equal contribution to the pole from I(u, s), and using the three-tachyon result (6.4.4), the condition (6.4.13) gives CD2 = e−λ CDX2 CDg 2 =

1 α go2

(6.4.14)

.

The three-tachyon amplitude is then 2igo (2π)26δ 26 ( i ki ) . (6.4.15) α The various functional determinants have dropped out. Using unitarity, all normalizations can be expressed in terms of the constant go appearing in the vertex operators. The determinants can in fact be computed by careful regularization and renormalization, and the relative normalizations of diﬀerent topologies agree with those from unitarity. Continuing past the pole, we encounter further singularities. Taylor expanding the integrand at y = 0 gives SD2 (k1 ; k2 ; k3 ) =

r

I(s, t) =

dy y −α s−2 + (α t + 2)y −α s−1 + . . .

.

(6.4.16)

0

The second term gives a pole at α s = 0, u−t I(s, t) = (6.4.17) + terms analytic at α s = 0 . 2s From the further terms in the Taylor expansion, the amplitude has poles at α s = −1, 0, 1, 2, . . . .

(6.4.18)

183

6.4 The Veneziano amplitude

These are precisely the positions of the open string states. The integral I(s, t) also has poles in the variable t at the same positions (6.4.18), coming from the endpoint y = 1. These are due to the process of ﬁgure 6.3(b). The other two terms in the amplitude (6.4.9) give further contributions to the s- and t-channel poles, and also give poles in the u-channel, ﬁgure 6.3(c). Because the residue at s = 0 in (6.4.17) is odd in u − t, this pole actually cancels that from I(s, u), as do all poles at even multiples of 1/α . This will not be true for the more general open string theories to be introduced in the next section. Deﬁne the Euler beta function 1

B(a, b) = 0

dy y a−1 (1 − y)b−1 ,

(6.4.19)

so that αo (x) = 1 + α x .

I(s, t) = B(−αo (s), −αo (t)) ,

(6.4.20)

This can be expressed in terms of gamma functions. Deﬁning y = v/w for ﬁxed w gives w

a+b−1

w

B(a, b) = 0

dv v a−1 (w − v)b−1 .

Multiplying both sides by e−w , integrating Γ(a + b)B(a, b) =

∞

dv v a−1 e−v

0

∞ 0

∞ 0

(6.4.21)

dw, and regrouping gives

d(w − v)(w − v)b−1 e−(w−v)

= Γ(a)Γ(b) .

(6.4.22)

The four-tachyon amplitude is then 2igo2 (2π)26δ 26 ( i ki ) α

× B(−αo (s), −αo (t)) + B(−αo (s), −αo (u)) + B(−αo (t), −αo (u)) ,

SD2 (k1 ; k2 ; k3 ; k4 ) =

(6.4.23) where Γ(−α x − 1)Γ(−α y − 1) . (6.4.24) Γ(−α x − α y − 2) This is the Veneziano amplitude, originally written to model certain features of strong interaction phenomenology. The high energy behavior of the Veneziano amplitude is important. There are two regions of interest, the Regge limit, B(−αo (x), −αo (y)) =

s→∞,

t ﬁxed ,

(6.4.25)

t/s ﬁxed .

(6.4.26)

and the hard scattering limit, s→∞,

184

6 Tree-level amplitudes

If we consider the scattering process 1 + 2 → 3 + 4 (so that k10 and k20 are positive and k30 and k40 negative), then in the 1-2 center-of-mass frame, θ θ , u = (4m2 − E 2 ) cos2 , (6.4.27) 2 2 where E is the center-of-mass energy and θ is the angle between particle 1 and particle 3. The Regge limit is high energy and small angle, while the hard scattering limit is high energy and ﬁxed angle. Using Stirling’s approximation, Γ(x+1) ≈ xx e−x (2π x)1/2 , the behavior in the Regge region is s = E2 ,

t = (4m2 − E 2 ) sin2

SD2 (k1 ; k2 ; k3 ; k4 ) ∝ sαo (t) Γ(−αo (t)) ,

(6.4.28)

where again αo (t) = α t + 1. That is, the amplitude varies as a power of s, the power being t-dependent. This is Regge behavior. At the poles of the gamma function, the amplitude is an integer power of s, corresponding to exchange of a string of integer spin αo (t). In the hard scattering limit, %

&

SD2 (k1 ; k2 ; k3 ; k4 ) ≈ exp −α (s ln sα + t ln tα + u ln uα ) = exp[−α sf(θ)] , (6.4.29) where θ θ θ θ f(θ) ≈ − sin2 ln sin2 − cos2 ln cos2 (6.4.30) 2 2 2 2 is positive. The result (6.4.29) is notable. High energy, ﬁxed angle scattering probes the internal structure of the objects being scattered. Rutherford discovered the atomic nucleus with hard alpha–atom scattering. Hard electron–nucleon scattering at SLAC revealed the quark constituents of the nucleon. In quantum ﬁeld theory, hard scattering amplitudes fall as a power of s. Even a composite object like the nucleon, if its constituents are pointlike, has power law amplitudes. The exponential falloﬀ (6.4.29) is very much softer. The result (6.4.29) suggests a smooth object of size α 1/2 , as one would expect. We started with the three-particle amplitude, skipping over the zero-, one-, and two-particle amplitudes. We will discuss these amplitudes, and their interpretation, in section 6.6. 6.5

Chan–Paton factors and gauge interactions

In this section we will consider the interactions of the massless vector state of the open string. To make the discussion a bit more interesting, we ﬁrst introduce a generalization of the open string theory. At the end of chapter 3 we introduced a very large class of bosonic string theories, but in this ﬁrst look at the interactions we are focusing

6.5 Chan–Paton factors and gauge interactions

185

j

i

Fig. 6.4. Open string with Chan–Paton degrees of freedom.

on the simplest case of 26 ﬂat dimensions. One can think of this in terms of symmetry: this theory has the maximal 26-dimensional Poincar´e invariance. In the closed bosonic string this is the unique theory with this symmetry. An outline of the proof is as follows. The world-sheet Noether currents for spacetime translations have components of weights (1, 0) and (0, 1). By an argument given in section 2.9, these currents are then holomorphic in z or ¯z . We have seen in the calculations in this chapter that this is enough to determine all the expectation values. In open string theory, however, there is a generalization. The open string has boundaries, endpoints. In quantum systems with distinguished points it is natural to have degrees of freedom residing at those points in addition to the ﬁelds propagating in the bulk. At each end of the open string let us add a new degree of freedom, known as a Chan–Paton degree of freedom, which can be in one of n states. A basis of string states is then |N; k; ij ,

(6.5.1)

where i and j denote the states of the left- and right-hand endpoints, running from 1 to n. The energy-momentum tensor is deﬁned to be the same as before, with no dependence on the new degrees of freedom. Conformal invariance is therefore automatic. Poincar´e invariance is automatic as well, as the Chan–Paton degrees of freedom are invariant. Although these new degrees of freedom have trivial world-sheet dynamics, they will have a profound eﬀect on the spacetime physics. In string theories of the strong interaction, the motivation for this was to introduce S U(3) ﬂavor quantum numbers: the endpoints are like quarks and antiquarks, connected by a color-electric ﬂux tube. Now we are motivating it in the general framework of considering all possibilities with given symmetries. We will give a new interpretation to the Chan– Paton degrees of freedom in chapter 8, and a possible further reﬁnement in chapter 14. There are now n2 scalar tachyons, n2 massless vector bosons, and so forth. The n2 Hermitian matrices λaij , normalized to Tr (λa λb ) = δ ab ,

(6.5.2)

are a complete set of states for the two endpoints. These are the representation matrices of U(n), so one might guess that the massless vectors are

186

6 Tree-level amplitudes

associated with a U(n) gauge symmetry; we will soon see that this is the case. Deﬁne the basis |N; k; a =

n

|N; k; ij λaij .

(6.5.3)

i,j=1

Now consider the four-tachyon amplitude shown in ﬁgure 6.2(a), in which the vertex operators are arranged in the cyclic order 1234. Because the Chan–Paton degrees of freedom do not appear in the Hamiltonian, their state does not evolve between the vertex operators: the right-hand endpoint of tachyon 1 must be in the same state as the left-hand endpoint of tachyon 2, and so forth. Thus, the amplitude 6.2(a) will now contain an additional factor of Tr (λa1 λa2 λa3 λa4 )

(6.5.4)

from the overlap of the Chan–Paton wavefunctions for each tachyon. This rule generalizes to an arbitrary amplitude: each vertex operator now contains a Chan–Paton factor λaij from the wavefunction of the endpoint degrees of freedom, and the amplitude for each world-sheet is multiplied by a trace of the Chan–Paton factors around each boundary. The three-tachyon amplitude becomes SD2 (k1 , a1 ; k2 , a2 ; k3 , a3 ) = igo (2π)26δ 26 ( i ki )Tr (λa1 λa2 λa3 + λa1 λa3 λa2 ) , (6.5.5) α the two cyclic orderings now having diﬀerent Chan–Paton traces. The four-tachyon amplitude is igo2 (2π)26δ 26 ( i ki ) α

× Tr (λa1 λa2 λa4 λa3 + λa1 λa3 λa4 λa2 )B(−αo (s), −αo (t))

SD2 (k1 , a1 ; k2 , a2 ; k3 , a3 ; k4 , a4 ) =

+ Tr (λa1 λa3 λa2 λa4 + λa1 λa4 λa2 λa3 )B(−αo (t), −αo (u))

+ Tr (λa1 λa2 λa3 λa4 + λa1 λa4 λa3 λa2 )B(−αo (s), −αo (u)) .

(6.5.6)

Considering again the unitarity relation (6.4.13), the pole at s = −1/α acquires a factor of 1 a1 a2 a3 a4 Tr {λ , λ }{λ , λ } 4 on the left and a factor of 1 a1 a2 a a3 a4 a Tr {λ , λ }λ Tr {λ , λ }λ , 4 a

(6.5.7)

(6.5.8)

187

6.5 Chan–Paton factors and gauge interactions

on the right, the sum being over the Chan–Paton wavefunction of the intermediate state. The completeness of the λaij and the normalization (6.5.2) imply that Tr (Aλa ) Tr (Bλa ) = Tr (AB)

(6.5.9)

for any matrices A and B, and so the amplitudes are still unitary. Gauge interactions The amplitude for a gauge boson and two tachyons is SD2 (k1 , a1 , e1 ; k2 , a2 ; k3 , a3 ) = −igo go2 e−λ e1µ ˙ µ eik1 ·X (y1 ) c1 eik2 ·X (y2 ) c1 eik3 ·X (y3 ) × c1 X

D2

Tr (λa1 λa2 λa3 )

+ (k2 , a2 ) ↔ (k3 , a3 ) . (6.5.10) We have used the gauge boson vertex operator (3.6.26), but for now are allowing an independent normalization constant go . Using the results of section 6.2, the X path integral is

˙ µ eik1 ·X (y1 ) eik2 ·X (y2 ) eik3 ·X (y3 ) X ×iCDX2 (2π)26δ 26 (

i ki )|y12 |

2α k1 ·k2

D2

= −2iα

µ

k3µ k2 + y12 y13

|y13 |2α k1 ·k3 |y23 |2α k2 ·k3 .

(6.5.11)

Using momentum conservation, the mass-shell conditions, and the physical state condition k1 ·e1 = 0, the amplitude becomes SD2 (k1 , a1 , e1 ; k2 , a2 ; k3 , a3 ) = −igo e1 ·k23 (2π)26δ 26 (

a1 a2 a3 i ki )Tr λ [λ , λ ] ,

(6.5.12)

where kij ≡ ki − kj . This is again independent of the vertex operator positions. The s = 0 pole in the four-tachyon amplitude no longer vanishes. The terms that canceled now have the Chan–Paton factors in diﬀerent orders, so the pole is proportional to

Tr [λa1 , λa2 ] [λa3 , λa4 ] .

(6.5.13)

Relating the coeﬃcient of this pole to the amplitude (6.5.12) by unitarity, one obtains go = (2α )−1/2 go .

(6.5.14)

This is the same relative normalization as from the state–operator mapping: there is only one independent coupling constant.

188

6 Tree-level amplitudes

For the three-gauge-boson coupling, a similar calculation gives SD2 (k1 , a1 , e1 ; k2 , a2 , e2 ; k3 , a3 , e3 ) = igo (2π)26δ 26 (

i ki ) e1 ·k23 e2 ·e3 + e2 ·k31 e3 ·e1 + e3 ·k12 e1 ·e2

α + e1 ·k23 e2 ·k31 e3 ·k12 Tr λa1 [λa2 , λa3 ] . 2

(6.5.15)

Up to ﬁrst order in the momenta, the amplitudes we have found are reproduced by the spacetime action 1 S = 2 go

1 1 21/2 d x − Tr (Dµ ϕDµ ϕ) + Tr (ϕ2 ) + 1/2 Tr (ϕ3 ) 2 2α 3α 1 − Tr (Fµν F µν ) , (6.5.16) 4 26

where the tachyon ﬁeld ϕ and the Yang–Mills vector potential Aµ are written as n × n matrices, e. g. Aµ = Aaµ λa . Also, Dµ ϕ = ∂µ ϕ − i[Aµ , ϕ] and Fµν = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ]. This is the action for a U(n) gauge ﬁeld coupled to a scalar in the adjoint representation. Adding in the Chan–Paton factors has produced just the gauge-invariant expressions needed. The gauge invariance is automatic because the decoupling of unphysical states is guaranteed in string perturbation theory, as we will develop further. At momenta k small compared to the string scale, the only open string states are the massless gauge bosons. As discussed in section 3.7, it is in this limit that the physics should reduce to an eﬀective ﬁeld theory of the massless states. We have therefore been somewhat illogical in including the tachyon in the action (6.5.16); we did so for illustration, but now let us focus on the gauge bosons. The four-gauge-boson amplitude has a form analogous to the Veneziano amplitude but with additional structure from the polarization tensors. Expanding it in powers of α k 2 , the ﬁrst term, which survives in the zero-slope limit α → 0, is the sum of pole terms in s, t, and u plus a constant. Consistency guarantees that it is precisely the four-gauge-boson amplitude obtained in ﬁeld theory from the Yang–Mills Lagrangian Fµν F µν . Note, however, the term of order α k 3 in the three-gauge-boson amplitude (6.5.15). This implies a higher derivative term 2iα − 2 Tr (Fµ ν Fν ω Fω µ ) (6.5.17) 3go in the Lagrangian. Similarly, expanding the four-point amplitude reveals an inﬁnite sum of higher order interactions (beyond (6.5.17), these do not contribute to the three-gauge-boson amplitude for kinematic reasons). String loop amplitudes also reduce, in the low energy region, to the loops

6.5 Chan–Paton factors and gauge interactions

189

obtained from the eﬀective Lagrangian. By the usual logic of eﬀective actions, the higher derivative terms are less important at low energy. The scale where they become important, α k 2 ≈ 1, is just where new physics (massive string states) appears and the eﬀective action is no longer applicable. If we have a cutoﬀ, why do we need renormalization theory? Renormalization theory still has content and in fact this is its real interpretation: it means that low energy physics is independent of the details of the high energy theory, except for the parameters in the eﬀective Lagrangian. This is a mixed blessing: it means that we can use ordinary quantum ﬁeld theory to make predictions at accelerator energies without knowing the form of the Planck scale theory, but it also means that we cannot probe the Planck scale theory with physics at particle accelerators. The string spectrum and amplitudes have an obvious global U(n) symmetry, λa → Uλa U † ,

(6.5.18)

which leaves the Chan–Paton traces and the norms of states invariant. From the detailed form of the amplitudes, we have learned that this is actually a local symmetry in spacetime. We will see that this promotion of a global world-sheet symmetry to a local spacetime symmetry is a rather general phenomenon in string theory. All open string states transform as the n × n adjoint representation under the U(n) symmetry. Incidentally, U(n) is not a simple Lie algebra: U(n) = S U(n) × U(1). The U(1) gauge bosons, λij = δij /n1/2 , decouple from the amplitudes (6.5.12) and (6.5.15). The adjoint representation of U(1) is trivial, so all string states are neutral under the U(1) symmetry.

The unoriented string It is interesting to generalize to the unoriented string. Consider ﬁrst the theory without Chan–Paton factors. In addition to M¨ obius invariance, the µ X CFT on the sphere or disk is invariant under the orientation-reversing symmetry σ 1 → π − σ 1 in the open string or σ 1 → 2π − σ 1 in the closed string. This world-sheet parity symmetry is generated by an operator Ω. From the mode expansions it follows that Ωαµn Ω−1 = (−1)n αµn

(6.5.19)

Ωαµn Ω−1 = ˜αµn

(6.5.20)

in the open string and

190

6 Tree-level amplitudes

in the closed string. The symmetry extends to the ghosts as well, but to avoid distraction we will not discuss them explicitly, since they only contribute a ﬁxed factor to the tree-level amplitudes. The tachyon vertex operators are even under world-sheet parity in either the closed or open string (this is obvious for the integrated vertex operators, without ghosts; the ﬁxed operators must then transform in the same way), determining the sign of the operator. All states can then be classiﬁed by their parity eigenvalue ω = ±1. The relation (6.5.19) in the open string implies that Ω|N; k = ωN |N; k ,

2

ωN = (−1)1+α m .

(6.5.21)

World-sheet parity is multiplicatively conserved. For example, the threetachyon amplitude is nonzero, consistent with (+1)3 = 1. On the other hand, the massless vector has ω = −1 and so we would expect the vector– tachyon–tachyon amplitude (6.5.10) and three-vector amplitude (6.5.15) to vanish in the absence of Chan–Paton degrees of freedom, as indeed they do (the λa are replaced by 1 and the commutators vanish). The diﬀerent cyclic orderings, related to one another by world-sheet parity, cancel. Given a consistent oriented string theory, we can make a new unoriented string theory by restricting the spectrum to the states of ω = +1. States with odd α m2 remain, while states with even α m2 , including the photon, are absent. The conservation of ω guarantees that if all external states have ω = +1, then the intermediate states in tree-level amplitudes will also have ω = +1. Unitarity of the unoriented theory thus follows from that of the oriented theory, at least at tree level (and in fact to all orders, as we will outline in chapter 9). The main point of interest in the unoriented theory is the treatment of the Chan–Paton factors. Since we have identiﬁed these with the respective endpoints of the open string, world-sheet parity must reverse them, Ω|N; k; ij = ωN |N; k; ji .

(6.5.22)

Again, this is a symmetry of all amplitudes in the oriented theory. To form the unoriented theory, we again restrict the spectrum to world-sheet parity eigenvalue ω = +1. Take a basis for λaij in which each matrix is either symmetric, sa = +1, or antisymmetric, sa = −1. Then Ω|N; k; a = ωN sa |N; k; a .

(6.5.23)

The world-sheet parity eigenvalue is ω = ωN sa , and the unoriented spectrum is α m2 even: λa antisymmetric , α m2 odd: λa symmetric .

(6.5.24a) (6.5.24b)

6.5 Chan–Paton factors and gauge interactions

191

For the massless gauge bosons the Chan–Paton factors are the n × n antisymmetric matrices and so the gauge group is S O(n). The states at even mass levels transform as the adjoint representation of the orthogonal group S O(n), and the states at odd mass levels transform as the traceless symmetric tensor plus singlet representations. The oriented theory has a larger set of orientation-reversing symmetries, obtained as a combination of Ω and a U(n) rotation, Ωγ |N; k; ij = ωN γjj |N; k; j i γi−1 i .

(6.5.25)

We can form more general unoriented theories by restricting the spectrum to ωγ = +1, which is again consistent with the interactions. Acting twice with Ωγ gives Ω2γ |N; k; ij = [(γ T )−1 γ]ii |N; k; i j (γ −1 γ T )j j .

(6.5.26)

We will insist that Ω2γ = 1 for reasons to be explained below. This then implies that γ T = ±γ .

(6.5.27)

That is, γ is symmetric or antisymmetric. A general change of Chan–Paton basis, |N; k; ij = Uii−1 |N; k; i j Uj j ,

(6.5.28)

γ = U T γU .

(6.5.29)

transforms γ to

In the symmetric case, it is always possible to ﬁnd a basis such that γ = 1, giving the theory already considered above. In the antisymmetric case there is a basis in which 0 I . (6.5.30) γ=M≡i −I 0 Here I is the k × k identity matrix and n = 2k must be even because γ is an invertible antisymmetric matrix. We take a basis for the Chan– Paton wavefunctions such that M(λa )T M = sa λa with sa = ±1. Then the world-sheet parity eigenvalue is ωγ = ωN sa , and the unoriented spectrum is α m2 even: M(λa )T M = −λa , α m2 odd: M(λa )T M = +λa .

(6.5.31a) (6.5.31b)

At the even mass levels, including the gauge bosons, this deﬁnes the adjoint representation of the symplectic group S p(k). The argument that we must have Ω2γ = 1 to construct the unoriented theory is as follows. Since Ω2 = 1, it must be that Ω2γ acts only on the

192

6 Tree-level amplitudes

Chan–Paton states, not the oscillators. In fact, from eq. (6.5.26), it acts on the Chan–Paton wavefunction as λ → (γ T )−1 γλγ −1 γ T = λ ,

(6.5.32)

where the last equality must hold in the unoriented theory, since all states in this theory are invariant under Ωγ and so also under Ω2γ . Now we assert that the allowed Chan–Paton wavefunctions must form a complete set. The point is that two open strings, by the splitting–joining interaction of ﬁgure 3.4(c), can exchange endpoints. In this way one can get to a complete set, that is, to any Chan–Paton state |ij (this argument is slightly heuristic, but true). By Schur’s lemma, if eq. (6.5.32) holds for a complete set, then γ −1 γ T = 1 and so Ω2γ = 1. We might try to obtain other gauge groups by taking diﬀerent sets of λa . In fact, the oriented U(n) and unoriented S O(n) and S p(k) theories constructed above are the only possibilities. Generalization of the completeness argument in eqs. (6.5.7)–(6.5.9) shows these to be the most general solutions of the unitarity conditions. In particular, exceptional Lie algebras, which are of interest in grand uniﬁcation and which will play a major role in volume two, cannot be obtained with Chan–Paton factors (in perturbation theory). In closed string theory there is another mechanism that gives rise to gauge bosons, and it allows other groups. 6.6

Closed string tree amplitudes

The discussion of closed string amplitudes is parallel to the above. The amplitude for three closed string tachyons is gc3 e−2λ

SS2 (k1 ; k2 ; k3 ) =

3

iki ·X

:˜cce

(zi , ¯zi ) :

.

i=1

(6.6.1)

S2

In this case the CKG P S L(2, C) (the M¨ obius group) can be used to ﬁx the three vertex operators to arbitrary positions z1,2,3 . Taking the expectation values from section 6.2, the result is again independent of the vertex operator positions, SS2 (k1 ; k2 ; k3 ) = igc3 CS2 (2π)26δ 26 (

i ki )

(6.6.2)

,

where CS2 = e−2λ CSX2 CSg2 . For four closed string tachyons,

SS2 (k1 ; k2 ; k3 ; k4 ) =

gc4 e−2λ

C

2

d z4

3 i=1

iki ·X

:˜cce

ik4 ·X

(zi , ¯zi ) : : e

(z4 , ¯z4 ) :

, S2

(6.6.3)

193

6.6 Closed string tree amplitudes

where the integral runs over the complex plane C. Evaluating the expectation value and setting z1 = 0, z2 = 1, z3 = ∞, this becomes SS2 (k1 ; k2 ; k3 ; k4 ) = igc4 CS2 (2π)26δ 26 ( where

J(s, t, u) = C

i ki )J(s, t, u)

(6.6.4)

,

d2 z4 |z4 |−α u/2−4 |1 − z4 |−α t/2−4 .

(6.6.5)

Here s + t + u = −16/α , but we indicate the dependence of J on all three variables to emphasize its symmetry among them. This amplitude converges when s, t, u < −4/α . It has poles in the variable u from z4 → 0, in the variable t from z4 → 1, and in the variable s from z4 → ∞. The poles are at the values α s , α t , α u = −4, 0, 4, 8, . . . ,

(6.6.6)

which are the masses-squared of the closed string states. The pole at α s = −4 is 8πig 4 CS 4 (6.6.7) igc CS2 d2 z4 |z4 |α s/2 ∼ − c 2 . αs+4 |z4 |>1/ Unitarity gives CS2 =

8π , α gc2

(6.6.8)

and so 8πigc (2π)26δ 26 ( i ki ) . (6.6.9) α Like the Veneziano amplitude, the amplitude for four closed string tachyons can be expressed in terms of gamma functions (exercise 6.10): SS2 (k1 ; k2 ; k3 ) =

8πigc2 (2π)26δ 26 ( α where αc (x) = 1 + α x/4 and SS2 (k1 ; k2 ; k3 ; k4 ) =

C(a, b) = C

= 2π

i ki )C(−αc (t), −αc (u))

,

(6.6.10)

d2 z |z|2a−2 |1 − z|2b−2 Γ(a)Γ(b)Γ(c) , Γ(a + b)Γ(a + c)Γ(b + c)

a + b + c = 1 . (6.6.11)

This is the Virasoro–Shapiro amplitude. There is just a single term, with poles in the s-, t-, and u-channels coming from the gamma functions in the numerator. Like the Veneziano amplitude, the Virasoro–Shapiro amplitude has Regge behavior in the Regge limit, SS2 (k1 ; k2 ; k3 ; k4 ) ∝ s2αc (t)

Γ(−αc (t)) , Γ(1 + αc (t))

(6.6.12)

194

6 Tree-level amplitudes

and exponential behavior in the hard scattering limit,

α SS2 (k1 ; k2 ; k3 ; k4 ) ∝ exp − (s ln sα + t ln tα + u ln uα ) . (6.6.13) 2 On the sphere, the amplitude for a massless closed string and two closed string tachyons is

¯ ν eik1 ·X (z1 , ¯z1 ) : SS2 (k1 , e1 ; k2 ; k3 ) = gc2 gc e−2λ e1µν :˜cc∂X µ ∂X :˜cceik2 ·X (z2 , ¯z2 ) : :˜cceik3 ·X (z3 , ¯z3 ) :

S2

πiα µ ν =− k23 (2π)26δ 26 ( i ki ) , (6.6.14) g e1µν k23 2 c where e1µν eµν 1 = 1. Expanding the Virasoro–Shapiro amplitude (6.6.10) on the s = 0 pole and using unitarity determines 2 gc = gc , (6.6.15) α again in agreement with the state–operator mapping with overall constant gc . The amplitude (6.6.14) would be obtained in ﬁeld theory from the spacetime action S + S T , where S is the action (3.7.20) for the massless ﬁelds, and where 1 4 2 ˜ 26 1/2 −2Φ µν ST = − d x (−G) e G ∂µ T ∂ν T − T , (6.6.16) 2 α is the action for the closed string tachyon T . For example, the amplitude for a graviton of polarization eµν is obtained from this action by expanding ˜ µν = ηµν − 2κeµν eik·x . G

(6.6.17)

Note that this is the Einstein metric, whose action (3.7.25) is independent of the dilaton. The normalization of the ﬂuctuation is determined by that of the graviton kinetic term in the spacetime action. Speciﬁcally, if one ˜ µν − ηµν = −2κeµν f(x) with eµν eµν = 1, the eﬀective action for takes G f has the canonical normalization 12 for a real scalar. The ﬁeld theory amplitude matches the string result (6.6.14) and relates the normalization of the vertex operators to the gravitational coupling, κ = πα gc = 2πgc .

(6.6.18)

The amplitude for three massless closed strings is iκ SS2 (k1 , e1 ; k2 , e2 ; k3 , e3 ) = (2π)26δ 26 ( i ki )e1µν e2αβ e3γδ T µαγ T νβδ , 2 (6.6.19) where α µ α γ µ αγ γ µα α γµ η + k31 η + k12 η + k23 k31 k12 . (6.6.20) T µαγ = k23 8

6.6 Closed string tree amplitudes

195

The order k 2 terms in this amplitude correspond to the spacetime action (3.7.25), while the k 4 and k 6 terms come from a variety of higher derivative interactions, including terms quadratic and cubic in the spacetime curvature. These higher corrections to the action can also be determined by calculating higher loop corrections to the world-sheet beta functions (3.7.14). The tensor structure of the closed string amplitude (6.6.19) is just two copies of that in the open string amplitude (6.5.15), if one sets α = 2 in the closed string and α = 12 in the open. The same holds for the amplitude (6.6.14). This is a consequence of the factorization of free-ﬁeld expectation values on the sphere into holomorphic and antiholomorphic parts. A similar factorization holds for four or more closed strings before integration over the vertex operator positions. Further, by careful treatment of the contour of integration it is possible to ﬁnd relations between the integrated amplitudes. For the four-tachyon amplitudes the integrals above are related J(s, t, u, α ) = −2 sin παc (t) I(s, t, 4α ) I(t, u, 4α ) ,

(6.6.21)

after use of the gamma function identity Γ(x)Γ(1 − x) sin(πx) = π; we now indicate the explicit dependence of the integrals on α . For the general integral appearing in four-point closed string amplitudes there is a relation

C

d2 z z a−1+m1 ¯z a−1+n1 (1 − z)b−1+m2 (1 − ¯z )b−1+n2 = 2 sin[π(b + n2 )]B(a + m1 , b + m2 )B(b + n2 , 1 − a − b − n1 − n2 ) . m1 − n1 ∈ Z , m2 − n2 ∈ Z (6.6.22)

This implies a corresponding relation between four-point open and closed string amplitudes πigc2 α sin[παc (t)]Ao (s, t, 14 α , go )Ao (t, u, 14 α , go )∗ , go4 (6.6.23) where the open string amplitudes include just one of the six cyclic permutations, with poles in the indicated channels. Ac (s, t, u, α , gc ) =

Consistency In chapter 9 we will discuss the convergence and gauge invariance of tree-level amplitudes in a general way, but as an introduction we will now use the OPE to see how these work for the lowest levels. Consider the operator product

¯ : eik1 ·X(z1 ,¯z1 ) : : eik4 ·X(z4 ,¯z4 ) : = |z14 |α k1 ·k4 : 1 + iz14 k1 ·∂X + i¯z14 k1 · ∂X

¯ + . . . ei(k1 +k4 )·X (z4 , ¯z4 ) : . − z14 ¯z14 k1 ·∂X k1 · ∂X

(6.6.24)

196

6 Tree-level amplitudes

This appears in the amplitude with z14 integrated. The integral converges as z14 → 0 when α (6.6.25) (k1 + k4 )2 − 4 > −2 2 and has a pole at that point. The coeﬃcient of the pole is just the tachyon vertex operator. Thus, if a pair of tachyons in any amplitude has total momentum (k1 + k4 )2 = 4/α , there will be a pole proportional to the amplitude with one fewer tachyon as required by unitarity. A pole in momentum space corresponds to long distance in spacetime, so this is a process in which two tachyons scatter into one, which then propagates and interacts with the remaining particles. Carrying the OPE further, the O(z14 ) and O(¯z14 ) terms do not produce poles because the angular integration gives zero residue, the O(z14 ¯z14 ) term gives a massless pole, and so on. Now let us look a little more closely at the way the local spacetime symmetries are maintained in the string amplitudes. The various amplitudes we have calculated all vanish if any of the polarizations are of the form eµ = kµ , or eµν = ζµ kν + kµ ˜ζν with k·ζ = k· ˜ζ = 0. The corresponding spacetime actions are thus invariant under the Yang–Mills, coordinate, and antisymmetric tensor symmetries. As discussed in section 3.6, the vertex operator for a longitudinal polarization is the sum of a total derivative and a term that vanishes by the equations of motion. Upon integration, the total derivative vanishes but the equation of motion term might have a source at one of the other vertex operators in the path integral. The operator product (6.6.24) vanishes rapidly when k1 · k4 is large. Using this property, there will be for any pair of vertex operators a kinematic region in which all possible contact terms are suppressed. The amplitude for any null polarization then vanishes identically in this region, and since all amplitudes are analytic except for poles (and branch points at higher order) the amplitudes for null polarizations must vanish everywhere. We see that the singularities required by unitarity, as well as any possible divergences or violations of spacetime gauge invariance, arise from the limits z → 0, 1, ∞, where two vertex operators come together. For the sphere with four marked points (vertex operators) these are the boundaries of moduli space. The analytic continuation argument used here is known for historic reasons as the canceled propagator argument. α k1 · k4 =

Closed strings on D2 and RP2 The lowest order closed–open interactions come from the disk with both closed and open vertex operators. The low energy eﬀective action for these can be deduced from general considerations. With a trivial closed string

6.6 Closed string tree amplitudes background we found the usual gauge kinetic term, 1 − 2 d26 x Tr (Fµν F µν ) . 4go

197

(6.6.26)

Obviously the metric must couple to this in a covariant way. In addition, the coupling of the dilaton can be deduced. Recall that go2 ∝ eΦ0 , with Φ0 the expectation value of the dilaton. So we should replace this (inside the ˜ The action integral) with Φ = Φ0 + Φ. 1 ˜ − 2 d26 x (−G)1/2 e−Φ Tr (Fµν F µν ) (6.6.27) 4go thus incorporates all interactions not involving derivatives of the closed string ﬁelds; indices are now raised and lowered with Gµν . This action reﬂects the general principle that the eﬀective action from Euler number χ is weighted by gc−χ ∝ e−χΦ . The disk and projective plane also make a contribution to purely closed string interactions. The amplitudes for n closed strings are of order go−2 gcn ∼ gcn−1 , one power of gc higher than the sphere. Closed string loop amplitudes, to be considered in the next chapter, are of order gc2 times the sphere, because emitting and reabsorbing a closed string adds two factors of gc . Thus the disk and projective plane are ‘half-loop order.’ Of particular interest are the amplitudes on the disk and the projective plane with a single closed string vertex operator. Fixing the position of the vertex operator removes only two of the three CKVs, the residual gauge symmetry consisting of rotations about the vertex operator position. Thus we have to divide the amplitude by the volume of this residual CKG. We have not shown how to do this explicitly, but will work it out for the torus in the next chapter. For the disk with one closed string this is a ﬁnite factor and the result is nonzero. The amplitude is a numerical factor times gc go−2 , which in turn is a pure number, times powers of α as required by dimensional analysis. We will not work out these numerical factors here, but will obtain them in an indirect way in chapter 8. Thus there is an amplitude for a single closed string to appear from the vacuum, either through the disk or the projective plane, necessarily with zero momentum. Such an amplitude is known as a tadpole. In other words, the background closed string ﬁelds are corrected from their original values at order gc . Again we can write an eﬀective action, which is simply −Λ

˜

d26 x (−G)1/2 e−Φ ,

(6.6.28)

the Φ dependence deduced as above. This is a potential for the dilaton. We will consider it further in the next chapter. The amplitude for a single closed string on the sphere, on the other hand, is zero. The residual CKG is a noncompact subgroup of P S L(2, C) and so

198

6 Tree-level amplitudes

one has to divide by an inﬁnite volume. A nonzero result would have been a logical inconsistency, a zeroth order correction to the background ﬁelds. Similarly the amplitude for two closed strings on the sphere (a zeroth order correction to the mass) vanishes, as do the corresponding disk amplitudes, one or two open strings. The amplitudes with no vertex operators at all ˜ 0 in the are also meaningful — they just calculate the term of order Φ Taylor expansion of the action (6.6.28). The disk amplitude with no vertex operators is thus nonvanishing; this requires a somewhat formal treatment of the conformal Killing volume. 6.7

General results

In this section we obtain some general results concerning CFT on the sphere and disk. M¨obius invariance We have seen that the sphere has a group of globally deﬁned conformal transformations, the M¨ obius group P S L(2, C), z =

αz + β γz + δ

(6.7.1)

for complex α, β, γ, δ with αδ − βγ = 1. This is the most general conformal transformation that is one-to-one on all of S2 , the complex z-plane plus the point at inﬁnity. Expectation values must be invariant under any M¨ obius transformation:

Ai (z1 , ¯z1 ) . . . Ak (zn , ¯zn ) S2 = Ai (z1 , ¯z1 ) . . . Ak (zn , ¯zn )

S2

.

(6.7.2)

We will consider the consequences of this symmetry for expectation values with one, two, three, or four local operators. For a single operator of weight (hi , ˜hi ), the rescaling plus rotation z = γz gives

Ai (0, 0) S2 = Ai (0, 0)

˜

S2

= γ −hi ¯γ −hi Ai (0, 0) S2 .

(6.7.3)

The one-point function therefore vanishes unless hi = ˜hi = 0. This is another way to see that the one-point string amplitude vanishes on the sphere, because the matter factor is the expectation value of a (1, 1) operator. For n = 2, we can use a translation plus z = γz to bring any pair of operators to the points 0 and 1, giving −h −hj −˜ h −˜ h ¯z12 i j

Ai (z1 , ¯z1 )Aj (z2 , ¯z2 ) S2 = z12 i

Ai (1, 1)Aj (0, 0) S2 ,

(6.7.4)

199

6.7 General results

so the position dependence is completely determined. Single-valuedness implies that Ji + Jj ∈ Z, where Ji = hi − ˜hi . There is a further constraint on the two-point function from the conformal transformation z = z + (z − z1 )(z − z2 ) + O(2 ), which leaves z1 and z2 ﬁxed. For general operators this is complicated, but for tensor ﬁelds Op and Oq it simply implies that

Op (z1 , ¯z1 )Oq (z2 , ¯z2 ) S2 = 0

unless hp = hq , ˜hp = ˜hq .

(6.7.5)

Any three points z1,2,3 can be brought to given positions by a M¨ obius transformation. For n ≥ 3, M¨ obius invariance therefore reduces the expectation value from a function of n complex variables to a function of n − 3 complex variables. Again the result takes a simple form only for tensor ﬁelds. For example, for three tensor ﬁelds one ﬁnds 3

Opi (zi , ¯zi )

i=1

= Cp1 p2 p3

3 h−2(˜ hi + ˜ h−2(hi +hj ) ˜ hj )

¯zij

zij

(6.7.6)

,

i,j=1 i<j

S2

where Cp1 p2 p3 is independent of position and h = h1 + h2 + h3 . For four primary ﬁelds, 4 i=1

˜

Opi (zi , ¯zi )

= Cp1 p2 p3 p4 (zc , ¯zc ) (z12 z34 )h (¯z12 ¯z34 )h ×

4 −hi −hj −˜ hi − ˜ hj

zij

i,j=1 i<j

S2

¯zij

,

(6.7.7)

hi , and zc = z12 z34 /z13 z24 is the M¨ where h = i hi , ˜h = i ˜ obius invariant cross-ratio. The function Cp1 p2 p3 p4 (zc , ¯zc ) is not determined by conformal invariance, so we are reduced from a function of four variables to an unknown function of one variable. On the disk represented as the upper half-plane, only the M¨ obius transformations with α, β, γ, δ real remain, forming the group P S L(2, R). The extension of the above is left as an exercise. One learns much more by considering the full conformal algebra. We will see in chapter 15 that it determines all expectation values in terms of those of the tensor ﬁelds. Path integrals and matrix elements The path integrals we have considered can be related to operator expressions. Consider the path integral on the sphere with two operators, one at the origin and one at inﬁnity:

Ai (∞, ∞)Aj (0, 0)

S2

.

(6.7.8)

The prime indicates the u-frame, which we have to take for the operator at inﬁnity; by a slight abuse of notation we still give the position in terms of z. Using the state–operator mapping we can replace the disk |z| < 1 containing Aj at z = 0 by the state ΨAj on the circle |z| = 1. We can

200

6 Tree-level amplitudes

also replace the disk |z| > 1 (|u| < 1) containing Ai at u = 0 by the state ΨAi on the circle |z| = 1. All that is left is the integral over the ﬁelds φb on the circle, so the expectation value (6.7.8) becomes

[dφb ]ΨAi [φΩ b ]ΨAj [φb ] ;

(6.7.9)

here φΩ b (σ) = φb (2π − σ), arising from the mapping zu = 1. This convolution of wavefunctions resembles an inner product, so we deﬁne

i|j =

Ai (∞, ∞)Aj (0, 0)

S2

(6.7.10)

.

This is essentially the inner product introduced by Zamolodchikov. For tensor operators in unitary theories, a M¨ obius transformation shows that this is the same as the coeﬃcient of 1 in the OPE, Oi (z, ¯z )Oj (0, 0) =

i|j

˜ ˜ z hi +hj ¯z hi +hj

1 S 2

+ ... .

(6.7.11)

We have abbreviated |Ai to |i . The

| product is not the same as the quantum mechanical inner product | introduced in chapter 4. The latter is Hermitean, whereas the former includes no complex conjugation and so is bilinear, up to a sign if i and j are anticommuting. That is,

i|j = ±

j|i ,

(6.7.12)

since all we have done is to interchange the two operators and rename z ↔ u. There is a simple relation between the two inner products, which we will develop later. We will sometimes write Gij for

i|j , and Gij for the inverse matrix, where i, j run over a complete set. The matrices Gij and Gij will be used to raise and lower indices, Ai = Gij Aj , Ai = Gij Aj . Operators in the path integral translate in the usual way into operators in Hilbert space. For example, ˆ k (1, 1)|j , A (∞, ∞)Ak (1, 1)Aj (0, 0) =

i|A (6.7.13) i

S2

where we reintroduce the hat to emphasize that we are in a Hilbert space formalism. Using the OPE, the left-hand side becomes

clkj Ai (∞, ∞)Al (0, 0)

S2

= cikj .

(6.7.14)

l

Thus the three-point expectation values on the sphere, the OPE coeﬃcients with all indices lowered, and the matrix elements of general local operators are all the same thing. By a M¨ obius transformation (rescaling of z) we have also

Ai (∞, ∞)Ak (z1 , ¯z1 )Aj (0, 0)

S2

˜ −˜ h −hk −hj h h −˜ h ¯z1 i k j cikj

= z1 i

.

(6.7.15)

201

6.7 General results

The four-point function translates into an operator expression ˆ k (z1 , ¯z1 )A ˆ l (z2 , ¯z2 ) ]|j , A (∞, ∞)Ak (z1 , ¯z1 )Al (z2 , ¯z2 )Aj (0, 0) =

i|T[ A i

S2

(6.7.16) where T denotes radial ordering. Let |z1 | > |z2 | and insert a complete set of states, 1 = |m Gmn

n| .

(6.7.17)

The four-point amplitude (6.7.16) becomes h −h −h ˜h −˜h −˜h hm −hl −hj ˜hm −˜hl −˜hj m m i i k k

¯z1

z1

¯z2

z2

cikm cmlj .

(6.7.18)

m

Thus, the operator product coeﬃcients determine not only the three-point expectation values on the sphere, but also the four-point and, by the same construction, arbitrary n-point amplitudes. For |z1 − z2 | > |z1 |, which overlaps the region |z1 | > |z2 | where the expansion (6.7.18) is valid, we can translate Ak to the origin and give a similar expansion in terms of cilm cmkj . The equality of these two expansions is associativity of the OPE, ﬁgure 2.8. Operator calculations The Hilbert space expressions give us one more way to calculate expectations values. We take as an example the case of four exponential operators

: eik4 ·X(∞,∞) : : eik1 ·X(z1 ,¯z1 ) : : eik2 ·X(z2 ,¯z2 ) : : eik3 ·X(0,0) : ik1 ·X1 ◦

=

0; k4 |T[ e ◦ ◦

◦ ◦ ◦

ik2 ·X2 ◦

e

◦

S2

]|0; k3 .

(6.7.19)

We have used the result (2.7.11) that : : ordered operators are the same as ◦◦ ◦◦ ordered operators for this CFT. We have abbreviated X µ (zi , ¯zi ) as Xiµ and to avoid clutter are omitting the hats on operators. By deﬁnition ◦ ◦

where

eik·X ◦◦ = eik·XC eik·XA ,

1/2 ∞ α 1

XCµ (z, ¯z )

= x −i

XAµ (z, ¯z )

α α = −i pµ ln |z|2 + i 2 2

µ

2

m m=1

(6.7.20)

(αµ−m z m + ˜αµ−m ¯z m ) , 1/2 ∞ 1 αµm

˜αµm + m z m ¯z m m=1

(6.7.21a)

.

(6.7.21b)

For |z1 | > |z2 | the matrix element (6.7.19) becomes

0; k4 |eik1 ·X1C eik1 ·X1A eik2 ·X2C eik2 ·X2A |0; k3 .

(6.7.22)

202

6 Tree-level amplitudes

To evaluate this use the Campbell–Baker–Hausdorﬀ (CBH) formula eik1 ·X1A eik2 ·X2C = eik2 ·X2C eik1 ·X1A e−[ k1 ·X1A , k2 ·X2C ] = eik2 ·X2C eik1 ·X1A |z12 |α k1 ·k2 .

(6.7.23)

The expectation value (6.7.22) becomes

|z12 |α k1 ·k2

0; k4 |eik1 ·X1C +ik2 ·X2C eik1 ·X1A +ik2 ·X2A |0; k3

= |z12 |α k1 ·k2

0; k4 |ei(k1 +k2 )·x eα (k1 ln |z1 |+k2 ln |z2 |)·p |0; k3

= |z12 |α k1 ·k2 |z1 |α k1 ·k3 |z2 |α k2 ·k3

0; k1 + k2 + k4 |0; k3

= iCSX2 (2π)d δ d (

i ki )|z12 |

α k1 ·k2

|z1 |α k1 ·k3 |z2 |α k2 ·k3 ,

(6.7.24)

where we have used the two-point expectation value to normalize the last line. This is the familiar result (6.2.31), obtained by two other methods in 2 section 6.2, after one includes in the latter a factor |z4 |α k4 from the change of frame and takes z4 → ∞. All other free-ﬁeld results can be obtained by this same oscillator method. Relation between inner products Nondegenerate bilinear and Hermitean inner products can always be related to one another by an appropriate antilinear operation on the bra. Let us consider an example. From the free-ﬁeld expectation values, we have

0; k|0; l = iCSX2 (2π)d δ d (k + l) ,

(6.7.25)

using the fact that |0; k maps to eik·X . Compare this with the inner product from the X µ CFT,

0; k|0; l = (2π)d δ d (l − k) .

(6.7.26)

These diﬀer only by k → −k from conjugating eik·X , and normalization,

0; k| = iCSX2 0; −k| .

(6.7.27)

For more general operators, there is a natural notion of conjugation in CFT. In Euclidean quantum mechanics, Hermitean conjugation inverts Euclidean time, eq. (A.1.37), so the natural operation of Euclidean conjugation is conjugation × time-reversal: an operator that is Hermitean in Minkowski space is also Hermitean under this combined operation. In CFT, we make the same deﬁnition, but also must include a time-reversal on the conformal frame, A(p) = A (p )† .

(6.7.28)

Here p and p are related by radial time-reversal, z = ¯z −1 , and the unprimed operator is in the z-frame and the primed operator in the u-

203

6.7 General results

frame. To see how this works, consider a holomorphic operator of weight h, whose Laurent expansion is O(z) = ih

∞

On . n+h n=−∞ z

(6.7.29)

Its simple adjoint is O(z)† = i−h

∞

O†n . z n+h n=−∞ ¯

(6.7.30)

Then the Euclidean adjoint is O(z) = i−h (−z −2 )h

∞ n=−∞

O†n

z −n−h

= ih

∞ O†−n n=−∞

z n+h

.

(6.7.31)

For an operator that is Hermitean in Minkowski time, O†−n = On , and so this operator is also Hermitean3 under O. The Euclidean adjoint (6.7.28) conjugates all explicit factors of i, but leaves z and ¯z indices unchanged. This is its whole eﬀect, other than an overall factor (−1)Na (Na −1)/2 from reversing the order of anticommuting ﬁelds; here Na is the total number of anticommuting ﬁelds in the operator. This is the natural conformally invariant operation of conjugation, so it must be that (6.7.32)

Ai | = K Ai | for some constant K. For a direct demonstration, the best we have come up with is

i | j ≡

1|Ai (∞, ∞)|j = K 1|Ai (∞, ∞)|j

= K 1|Ai (0, 0)† |j = K j|Ai (0, 0)|1 ∗ = K j|i ∗ = K i|j .

(6.7.33)

The only step here that is not either a deﬁnition or obvious is the assumption of proportionality,

1| = K 1|. This must hold because |1 is the unique S L(2,C)-invariant state. For the X CFT we have seen that K X = iCSX2 . For the ghost CFT, the Laurent expansion of the amplitude (6.3.4) gives

0|˜c0 c0 |0 = −CSg2 , where |0 = ˜c1 c1 |1 . The Hermitean inner product was deﬁned in eq. (4.3.18) as

0|˜c0 c0 |0 = i, so K g = iCSg2 . Except for the is, which are absent in a unitary CFT, one can set K = 1 by adding an Euler number term to the action. In a Hermitean basis of operators, the two inner products are then identical, and the distinction 3

Because its Laurent expansion has no factor of i, the Faddeev–Popov c ghost is actually anti-Hermitean under the Euclidean adjoint. This is due to an inconvenient conﬂict between conventions, and has no signiﬁcance.

204

6 Tree-level amplitudes

between them is often ignored. Notice, however, that a vertex operator of deﬁnite nonzero momentum cannot be Hermitean. Again, this can all be extended to the open string, with the disk in place of the sphere. Exercises 6.1 Verify that the expectation value (6.2.31) is smooth in the u-patch. 6.2 For the linear dilaton CFT, Φ = Vµ X µ , the X µ zero-mode path integral diverges. This reﬂects the fact that the coupling is diverging in some direction. Consider therefore imaginary Φ = ibµ X µ ; this is unphysical but has technical applications (one could instead take complex momenta). (a) Generalize the calculation that gave (6.2.17) to this case. Show that in the end the only changes are in the Weyl dependence (reﬂecting a change in the dimension of the exponential) and in the spacetime momentum conservation. (b) Generalize the calculation of the same quantity using holomorphicity. Show that the result is the same as in the regular X µ CFT but with an extra background charge operator exp(−2ib · X) taken to inﬁnity. 6.3 For one b and four c ﬁelds, show that expressions (6.3.5) and (6.3.8) are equal. Ignore the antiholomorphic ﬁelds. 6.4 (a) Write the amplitude for n open string tachyons on the disk as a generalization of eq. (6.4.5). (b) Show by a change of variables that it is independent of the positions of the ﬁxed vertex operators. 6.5 (a) Show that the residue of the pole in I(s, t) at α s = J − 1 is a polynomial in u − t of degree J, corresponding to intermediate particles of spin up to J. (b) Consider the amplitude I(s, t) without assuming D = 26. In the centerof-mass frame, write the residue of the pole at α s = 1 in terms of J k3k k3l , k1i k1j Pij,kl

where P 0 projects onto spin 0 (the unit matrix) and P 2 projects onto spin 2 (traceless symmetric tensors). Show that the coeﬃcient of P 0 is positive for D < 26, zero at D = 26, and negative at D > 26. Compare with the expected string spectrum at this level, and with the discussion of this level in section 4.1. 6.6 (a) In the hard scattering limit, evaluate the integral in the Veneziano amplitude (6.4.5) using the saddle point approximation. Take the naive saddle point; the justiﬁcation requires consideration of the analytic continuations needed to deﬁne the integral. (b) Do the same for the Virasoro–Shapiro amplitude.

Exercises

205

6.7 (a) Derive the amplitude (6.5.12). (b) Verify the relation (6.5.14) between vertex operator normalizations. Show that this agrees with the state–operator mapping. 6.8 Derive the three-gauge-boson amplitude (6.5.15). 6.9 (a) Obtain the four-gauge-boson amplitude. To reduce the rather extensive algebra, consider only those terms in which the polarizations appear as e1 ·e2 e3 ·e4 . (b) Compare the behavior at small α ki · kj with the same amplitude in Yang–Mills theory. 6.10 Carry out the integral in eq. (6.6.5) to obtain the Virasoro–Shapiro amplitude (6.6.10). The relation |z|

−2a

1 = Γ(a)

∞

dt ta−1 exp(−tz¯z )

0

is useful. (Reference: Green, Schwarz, & Witten (1987), section 7.2; note that their d2 z is 12 times ours.) 6.11 (a) Verify the amplitude (6.6.14). (b) Verify the relation (6.6.15) between vertex operator normalizations. (c) Complete the ﬁeld theory calculation and verify the relation (6.6.18) between gc and the gravitational coupling. 6.12 Calculate the disk amplitude with one closed string tachyon and two open string tachyons. 6.13 (a) Find the P S L(2, C) transformation that takes three given points z1,2,3 into chosen positions zˆ 1,2,3 . (b) Verify the M¨ obius invariance results (6.7.3)–(6.7.7). Show that to derive ˜ 1 annihilate the operators. (6.7.5) it is suﬃcient that L1 and L 6.14 (a) Find the P S L(2, R) transformation that takes three given points y1,2,3 into chosen positions yˆ 1,2,3 . (b) Generalize the M¨ obius invariance results (6.7.3)–(6.7.7) to operators on the boundary of a disk. (c) Find similar results for the disk with one interior, two interior, and one interior and one boundary operators. 6.15 (a) Generalize the operator calculation (6.7.24) to a product of n exponentials. (b) Generalize it to the expectation value (6.2.25).

7 One-loop amplitudes

After describing the relevant surfaces, we focus on the torus, ﬁrst on the CFTs on this surface and then on the string amplitudes. We then generalize to the open and unoriented theories. The most important issue to be understood is the absence of short-distance (UV) divergences. 7.1

Riemann surfaces

There are four Riemann surfaces with Euler number zero. The torus The torus T 2 , discussed in section 5.1, is the only closed oriented surface with Euler number zero. We describe it as the complex plane with metric ¯ and identiﬁcations ds2 = dwdw w∼ (7.1.1) = w + 2π ∼ = w + 2πτ . There are two moduli, the real and imaginary parts of τ = τ1 + iτ2 , and two CKVs, the translations. In terms of the real coordinates w = σ 1 + iσ 2 , (σ 1 , σ 2 ) ∼ (7.1.2) = (σ 1 + 2π, σ 2 ) ∼ = (σ 1 + 2πτ1 , σ 2 + 2πτ2 ) , so that one can think of the torus as a cylinder of circumference 2π and length 2πτ2 with the ends rotated by an angle of 2πτ1 and then sewn together. In terms of the coordinate z = exp(−iw), the identiﬁcation w ∼ = w + 2π is automatic, while w ∼ = w + 2πτ becomes z∼ (7.1.3) = z exp(−2πiτ) . A fundamental region is the annulus 1 ≤ |z| ≤ exp(2πτ2 ) . 206

(7.1.4)

7.1 Riemann surfaces

207

The torus is formed by rotating the outer circle by 2πτ1 and then sewing the inner and outer circles together. We will use the w-coordinate unless otherwise noted. The cylinder (annulus) The cylinder C2 is the region 0 ≤ Re w ≤ π ,

w∼ = w + 2πit .

(7.1.5)

That is, it is a strip of width π and length 2πt, with the ends joined. There is a single modulus t, which runs over the whole range 0 < t < ∞. Unlike the torus there is no modular group, the long-cylinder limit t → 0 being quite diﬀerent from the long-strip limit t → ∞. There is a single CKV, the translation parallel to the boundary. The cylinder can be obtained from the torus with imaginary τ = it by identifying under the involution ¯ , w = −w

(7.1.6)

which is a reﬂection through the imaginary axis. The lines σ 1 = 0, π are ﬁxed by this reﬂection and so become boundaries. The other coordinate is periodically identiﬁed, (σ 1 , 0) ∼ = (σ 1 , 2πt) .

(7.1.7)

The Klein bottle The Klein bottle K2 can be regarded as the complex plane with identiﬁcation ¯ + 2πit w∼ = w + 2π ∼ = −w

(7.1.8)

(σ 1 , σ 2 ) ∼ = (σ 1 + 2π, σ 2 ) ∼ = (−σ 1 , σ 2 + 2πt) .

(7.1.9)

or

This is a cylinder of circumference 2π and length 2πt, with the ends sewn together after a parity-reversal Ω. The single modulus t runs over 0 < t < ∞, and there is no modular group. Translation in the σ 2 direction is the only CKV. The Klein bottle can be obtained from the torus with modulus τ = 2it by identifying under ¯ + 2πit . w = −w

(7.1.10)

The Klein bottle can be thought of as a sphere with two cross-caps, but we postpone this description until section 7.4.

208

7 One-loop amplitudes The M¨obius strip

The M¨ obius strip M2 is sewn from a strip with a twist by Ω, 0 ≤ Re w ≤ π ,

¯ + π + 2πit . w∼ = −w

(7.1.11)

Again the modulus t runs over 0 < t < ∞ and the only CKV is the σ 2 translation. It can be obtained from the torus with τ = 2it by identifying under the two involutions ¯ w = −w

and

w = w + π(2it + 1) .

(7.1.12)

The M¨ obius strip can be thought of as a disk with a cross-cap, but again we defer this description. 7.2

CFT on the torus Scalar correlators

As in the case of the sphere, we start with the Green’s function (6.2.7), satisfying 2¯ 1 ¯ w, w ¯ ) = −2πδ 2 (w − w ) + ∂∂G (w, w; . α 4πτ2

(7.2.1)

The Green’s function is periodic in both directions on the torus, and aside from the source and the background charge term it would be the sum of holomorphic and antiholomorphic functions. These properties identify it as being associated with the theta functions, whose properties are given at the end of this section. In particular we guess ¯ w, w ¯ ) ∼ − G (w, w;

w − w 2 α ln ϑ1 τ . 2 2π

(7.2.2)

The function ϑ1 vanishes linearly when its ﬁrst argument goes to zero, giving the correct behavior as w → w . Elsewhere the logarithm is the sum of holomorphic and antiholomorphic functions and so is annihilated ¯ However, this function is not quite doubly periodic owing to the by ∂∂. quasiperiodicity (7.2.32b) — it changes by −α [Im(w − w ) + πτ2 ] under w → w + 2πτ. Also, the background charge is missing. Both properties are easily corrected:

w − w 2 [Im(w − w )]2 α + k(τ, ¯τ) . ln ϑ1 τ + α 2 2π 4πτ2 (7.2.3) The function k(τ, ¯τ) is determined by orthogonality to X0 , but as in the case of the sphere it drops out due to spacetime momentum conservation. ¯ w, w ¯ ) = − G (w, w;

209

7.2 CFT on the torus

The expectation value of a product of vertex operators is given, in parallel to the earlier result (6.2.13) on the sphere, by n i=1

iki ·X(zi ,¯zi )

:e

= iCTX2 (τ)(2π)d δ d (

:

i ki )

T2

α ki ·kj 2π wij (Im wij )2 . × ϑ τ exp − ∂ν ϑ1 (0|τ) 1 2π 4πτ2

(7.2.4)

i<j

The factor 2π/∂ν ϑ1 comes from the renormalized self-contractions. The scalar partition function The overall normalization of the sphere was absorbed into the string coupling constant, but we cannot do this for the torus, in particular because there is a nontrivial τ-dependence. In fact, we will see that there is a great deal of physics in the amplitude with no vertex operators at all. Consider the path integral with no vertex operators, 1 T 2 (τ) ≡ Z(τ). We can think of the torus with modulus τ as formed by taking a ﬁeld theory on a circle, evolving for Euclidean time 2πτ2 , translating in σ 1 by 2πτ1 , and then identifying the ends. In operator language this gives a trace,

Z(τ) = Tr exp(2πiτ1 P − 2πτ2 H)

˜

= (q¯ q )−d/24 Tr q L0 q¯L0

(7.2.5)

.

˜ 0 generates translations of Here q = exp(2πiτ), the momentum P = L0 − L 1 1 ˜ σ , and the Hamiltonian H = L0 + L0 − 24 (c + ˜c) generates translations of σ 2 as in eq. (2.6.10). Such a trace, weighted by the exponential of the Hamiltonian and other conserved quantities, is termed a partition function as in statistical mechanics. The trace breaks up into a sum over occupation numbers Nµn and ˜ µn for each µ and n and an integral over momentum k µ . The Virasoro N generators similarly break up into a sum, and Z(τ) becomes q )−d/24 Vd (q¯

dd k exp(−πτ2 α k 2 ) d (2π) µ,n

∞

˜

q nNµn q¯nNµn .

(7.2.6)

˜ µn =0 Nµn ,N

normalizaThe factor of spacetime volume Vd comes from the continuum −d d d k. The various sums tion of the momentum, k becoming Vd (2π) are geometric, ∞ N=0

q nN = (1 − q n )−1 ,

(7.2.7)

210

7 One-loop amplitudes

and so we obtain Z(τ) = iVd ZX(τ)d ,

(7.2.8)

ZX(τ) = (4π 2 α τ2 )−1/2 |η(τ)|−2 .

(7.2.9)

where

Here η(τ) = q 1/24

∞

(1 − q n )

(7.2.10)

n=1

is the Dedekind eta function, discussed further at the end of the section. The i comes from the rotation k 0 → ik d needed to deﬁne the divergent integral. Also, the earlier CTX2 (τ) is ZX(τ)d . The result (7.2.8) is obviously invariant under τ → τ + 1. Invariance of ZX under τ → −1/τ follows from eq. (7.2.44), η(−1/τ) = (−iτ)1/2 η(τ), thus generating the full modular group. Similarly the expectation value with vertex operators is modular-covariant, taking into account the Weyl transformations of the operators. We have evaluated the path integral by translating into an operator expression, but as on the sphere there are two other methods. We leave to the reader the direct path integral evaluation. Fourier transforming in σ 1 , it just breaks up into a product of harmonic oscillator path integrals. The evaluation using holomorphicity is less direct. We include it for illustration because it is used at higher genus and for more complicated CFTs. The idea is to get a diﬀerential equation with respect to τ. Consider the torus with modulus τ and make a small change in the metric, δgww = ∗ . The new metric is ¯ + ∗ dw 2 + dw ¯2 ds2 = dwdw ¯ − w)] d[w ¯ + ∗ (w − w)] ¯ + O(2 ) . (7.2.11) = (1 + ∗ + ) d[w + (w ¯ with w = w + (w ¯ − w). That is, the metric is Weyl-equivalent to dw dw This has the periodicities w ∼ (7.2.12) = w + 2π(τ − 2iτ2 ) . = w + 2π ∼ This change in the metric is therefore equivalent to a change δτ = −2iτ2

(7.2.13)

in the modulus. Under a change in the metric, the change in the path integral is

1 ¯

d2 w δgw¯ w¯ Tww (w) + δgww Tw¯ w¯ (w) δZ(τ) = − 2π

= −2πi δτ Tww (0) − δ¯τ Tw¯ w¯ (0) .

(7.2.14)

211

7.2 CFT on the torus

In the second line we have used translation invariance to carry out the integral over w. All expectation values are understood to be on the torus with modulus τ. To get the expectation value of the energy-momentum tensor, we use the OPE ∂w X µ (w)∂w Xµ (0) = −

α d − α Tww (0) + O(w) . 2w 2

(7.2.15)

Now, ¯ w, w ¯ )|w =0 Z(τ)−1 ∂w X µ (w)∂w Xµ (0) = d∂w ∂w G (w, w; =

α d ϑ1 ∂w2 ϑ1 − ∂w ϑ1 ∂w ϑ1 α d + , 2 8πτ2 ϑ21

(7.2.16)

where the arguments of all the theta functions are (w/2π, τ). This does indeed have a double pole at w = 0. Carefully expanding both numerator and denominator to order w 2 (this is simpliﬁed by the fact that ϑ1 is odd), the order w 0 term in the expectation value (7.2.16) is α d α d ∂w3 ϑ1 + . 6 ∂w ϑ1 8πτ2

(7.2.17)

The OPE (7.2.15) then gives

d ∂3 ϑ1 (0|τ) d

Tww (0) = − w − 6 ∂w ϑ1 (0|τ) 8πτ2

Z(τ) ,

(7.2.18)

and the variation (7.2.14) gives the diﬀerential equation ∂τ ln Z(τ) =

πid ∂w3 ϑ1 (0|τ) id . + 3 ∂w ϑ1 (0|τ) 4τ2

(7.2.19)

To proceed further, we use the fact, which is easily veriﬁed from the inﬁnite sum, that the theta functions satisfy

∂w2 ϑ1

w w i τ = ∂τ ϑ1 τ 2π π 2π

.

(7.2.20)

Rewrite (7.2.19) as d id ∂τ ln Z(τ) = − ∂τ ln ∂w ϑ1 (0|τ) + . 3 4τ2

(7.2.21)

Together with the conjugate equation, this determines −d/2

Z(τ) = |∂w ϑ1 (0|τ)|−2d/3 τ2

(7.2.22)

up to a numerical coeﬃcient that must be ﬁxed by other means. Noting the relation (7.2.43), this is the same as the earlier result (7.2.8).

212

7 One-loop amplitudes The bc CFT

For the ghost CFT, the partition function is again determined easily in terms of a trace over states. There are two sets of raising operators b−n and c−n on both the right and left, but now these are anticommuting so the occupation numbers can only be zero or one — a single oscillator at mode n thus gives (1 + q n ). In addition there are four ground states, giving

˜

Tr exp(2πiτ1 P − 2πτ2 H) = (q¯ q )13/12 Tr q L0 q¯L0 = 4(q¯ q )1/12

∞

|1 + q n |4 .

(7.2.23)

n=1

Recall, however, that for anticommuting ﬁelds the simple trace corresponds to a path integral with antiperiodic boundary conditions in the time direction. In order to calculate the Faddeev–Popov determinant, the ghosts must have the same periodicity as the original coordinate transformations, which were periodic. Thus we need

Z(τ) = Tr (−1)F exp(2πiτ1 P − 2πτ2 H)

=0,

(7.2.24)

where (−1)F anticommutes with all the ghost ﬁelds. The trace vanishes because the states | ↓↓ and | ↑↑ have opposite ghost number mod 2 from | ↑↓ and | ↓↑ . From the path integral point of view, Z(τ) vanishes due to ghost zero modes from the moduli and CKVs. These must be saturated with appropriate insertions, the simplest nonvanishing amplitude (and the one we will need) being

¯ 3 )˜b(w ¯ 4) c(w1 )b(w2 )˜c(w

(7.2.25)

.

In the operator calculation we write each ﬁeld in terms of its mode expansion. Only the n = 0 terms contribute, the expectation values of the others vanishing as above. Then (7.2.25) becomes

Tr (−1)F c0 b0˜c0 ˜b0 exp(2πiτ1 P − 2πτ2 H) .

(7.2.26)

b0 projects onto the single ground state | ↑↑ , so the The operator c0 b0˜c0 ˜ result is (q¯ q )1/12

∞

|1 − q n |4 = |η(τ)|4 .

(7.2.27)

n=1

Note the sign change in the inﬁnite product, from the (−1)F in the trace. The result (7.2.27) is independent of the positions of the ghost ﬁelds because the CKV and quadratic diﬀerentials on the torus are constants.

7.2 CFT on the torus

213

Again, we do not try to keep track of the overall sign, setting it to be positive.

General CFTs For general CFTs, Z(τ) =

˜

q hi −c/24 q¯hi −˜c/24 (−1)Fi ,

(7.2.28)

i

where i runs over all states of the CFT, and Fi is again the world-sheet 1 (c − ˜c) fermion number. Invariance under τ → τ + 1 requires hi − ˜hi − 24 to be an integer. Looking at the unit operator, this requires c − ˜c to be a multiple of 24. For general operators one then has the requirement that the spin be an integer, hi − ˜hi ∈ Z .

(7.2.29)

Incidentally, CFTs for which c − ˜c is not a multiple of 24 are certainly of interest. These cannot be separately modular-invariant, but with appropriate projections can be combined into a modular-invariant theory. Invariance under τ → −1/τ puts a further constraint on the spectrum. Let us here note just one general consequence, which relates the states of large h to those of lowest h. Consider the partition function for τ = i as → 0. The convergence factor q = exp(−2π) is approaching unity, so the partition function is determined by the density of states at high weight. Using modular invariance, this is equal to the partition function at τ = i/. In the latter case, q = exp(−2π/) is going to zero, so the state of ˜ 0 = 0, lowest weight dominates the sum. This is the unit state, with L0 = L giving →0

Z(i) ≈ exp

π(c + ˜c) . 12

(7.2.30)

The density of states at high weight is thus determined by the central charge. This generalizes the free-boson result, where c = ˜c = d counts the number of free bosons.1

1

To be precise, the result (7.2.30) holds in compact unitary CFTs. In nonunitary theories there are usually operators of negative weight, whose contribution will dominate that of the unit operator. In the linear dilaton theory, the central charge depends on V µ but the density of states does not: here the exception occurs because of complications due to the noncompactness of X µ combined with lack of translation invariance.

214

7 One-loop amplitudes Theta functions

The basic theta function is ∞

ϑ(ν, τ) =

exp(πin2 τ + 2πinν) .

(7.2.31)

n=−∞

It has the periodicity properties ϑ(ν + 1, τ) = ϑ(ν, τ) , ϑ(ν + τ, τ) = exp(−πiτ − 2πiν)ϑ(ν, τ) ,

(7.2.32a) (7.2.32b)

and these in fact determine it up to normalization. Under modular transformations, ϑ(ν, τ + 1) = ϑ(ν + 1/2, τ) , ϑ(ν/τ, −1/τ) = (−iτ)

1/2

(7.2.33a) 2

exp(πiν /τ)ϑ(ν, τ) .

(7.2.33b)

The theta function has a unique zero, up to the periodicity (7.2.32), at ν = 12 (1 + τ). It can also be written as an inﬁnite product, ϑ(ν, τ) =

∞

(1 − q m )(1 + zq m−1/2 )(1 + z −1 q m−1/2 ) ,

(7.2.34)

m=1

in terms of q = exp(2πiτ),

z = exp(2πiν) .

(7.2.35)

We often need the asymptotic behavior of the theta function as q → 0 or as q → 1. The q → 0 behavior can be read immediately from either the inﬁnite sum or inﬁnite product form. The q → 1 behavior is not manifest in either form, because an inﬁnite number of terms contribute. It can be obtained from the τ → −1/τ modular transformation, which relates these two limits by q → exp(4π 2 / ln q). It is also useful to deﬁne the theta function with characteristics,

ϑ

a (ν, τ) = exp πia2 τ + 2πia(ν + b) ϑ(ν + aτ + b, τ) b

=

∞

exp πi(n + a)2 τ + 2πi(n + a)(ν + b) .

(7.2.36)

n=−∞

Other common notations are

ϑ00 (ν, τ) = ϑ3 (ν|τ) = ϑ

ϑ01 (ν, τ) = ϑ4 (ν|τ) = ϑ

∞

2 0 (ν, τ) = q n /2 z n , 0 n=−∞

(7.2.37a)

∞

2 0 (ν, τ) = (−1)n q n /2 z n , 1/2 n=−∞

(7.2.37b)

215

7.2 CFT on the torus

∞

2 1/2 (ν, τ) = q (n−1/2) /2 z n−1/2 , ϑ10 (ν, τ) = ϑ2 (ν|τ) = ϑ 0 n=−∞ (7.2.37c) 1/2 (ν, τ) ϑ11 (ν, τ) = −ϑ1 (ν|τ) = ϑ 1/2 ∞

= −i

(−1)n q (n−1/2)

2 /2

z n−1/2 .

(7.2.37d)

n=−∞

These have the product representations ϑ00 (ν, τ) = ϑ01 (ν, τ) =

∞

(1 − q m )(1 + zq m−1/2 )(1 + z −1 q m−1/2 ) ,

m=1 ∞

(1 − q m )(1 − zq m−1/2 )(1 − z −1 q m−1/2 ) ,

(7.2.38a) (7.2.38b)

m=1

ϑ10 (ν, τ) = 2 exp(πiτ/4) cos πν

∞

(1 − q m )(1 + zq m )(1 + z −1 q m ) ,

m=1

ϑ11 (ν, τ) = −2 exp(πiτ/4) sin πν

(7.2.38c)

∞

(1 − q m )(1 − zq m )(1 − z −1 q m ) .

m=1

(7.2.38d)

Their modular transformations are ϑ00 (ν, τ + 1) ϑ01 (ν, τ + 1) ϑ10 (ν, τ + 1) ϑ11 (ν, τ + 1)

= = = =

ϑ01 (ν, τ) , ϑ00 (ν, τ) , exp(πi/4)ϑ10 (ν, τ) , exp(πi/4)ϑ11 (ν, τ) ,

(7.2.39a) (7.2.39b) (7.2.39c) (7.2.39d)

and ϑ00 (ν/τ, −1/τ) = (−iτ)1/2 exp(πiν 2 /τ)ϑ00 (ν, τ) ,

(7.2.40a)

ϑ01 (ν/τ, −1/τ) = (−iτ)1/2 exp(πiν 2 /τ)ϑ10 (ν, τ) ,

(7.2.40b)

ϑ10 (ν/τ, −1/τ) = (−iτ)

(7.2.40c)

1/2

ϑ11 (ν/τ, −1/τ) = −i(−iτ)

2

exp(πiν /τ)ϑ01 (ν, τ) ,

1/2

2

exp(πiν /τ)ϑ11 (ν, τ) .

(7.2.40d)

We will encounter these functions more extensively in the superstring. The theta function satisﬁes Jacobi’s ‘abstruse identity,’ a special case of a quartic identity of Riemann, ϑ400 (0, τ) − ϑ401 (0, τ) − ϑ410 (0, τ) = 0 .

(7.2.41)

ϑ11 (0, τ) = 0 .

(7.2.42)

Note also that

216

7 One-loop amplitudes

Finally, the Dedekind eta function is η(τ) = q

1/24

∞

∂ν ϑ11 (0, τ) (1 − q ) = −2π m=1 m

1/3

(7.2.43)

.

It has the modular transformations η(τ + 1) = exp(iπ/12) η(τ) , η(−1/τ) = (−iτ) 7.3

1/2

(7.2.44a) (7.2.44b)

η(τ) .

The torus amplitude

We now apply to the torus the general result (5.3.9) expressing the string amplitude as an integral with ghost and vertex operator insertions. The two CKVs require one vertex operator to be ﬁxed, so ST 2 (1; 2; . . . ; n) 1 = 2

dτd¯τ F0

˜ ˜ccV1 (w1 , w ¯ 1) BB

n i=2

¯ i Vi (wi , w ¯ i) dwi dw

.

(7.3.1)

T2

The fundamental region F0 and the factor of 12 from w → −w were discussed in section 5.1. As usual for complex variables, dτd¯τ = 2dτ1 dτ2 , where τ = τ1 + iτ2 . The ghost insertion for dτ is 1 1 d2 w bww (w)∂τ gw¯ w¯ B= (b, ∂τ g) = 4π 2π i = d2 w bww (w) 4πτ2 → 2πibww (0) . (7.3.2) We have used the result (7.2.13) for the τ-derivative of the metric. In the ﬁnal line we have used the fact that the ghost path integral (7.2.27) is independent of position to carry out the w-integration. (Recall that d2 w = 2dσ 1 dσ 2 and that gww¯ = 12 .) The CKG consists of the translations of the torus. Since this group has ﬁnite volume, we need not ﬁx it: we can rewrite the amplitude so that all vertex operators are integrated and the volume of the CKG is divided out. It is easy to do this by hand here. The CKVs are constant, so the expectation value in eq. (7.3.1) is independent of where the c ghosts are placed: we can move them away from w1 , putting them at some ﬁxed position. For the vertex operators, translation invariance implies that the amplitude is unchanged if we replace wi → wi + w. Average over translations ¯ dwdw , (7.3.3) 2 2(2π) τ2

217

7.3 The torus amplitude

the denominator being the area of the torus, which is the volume of the CKG. Using eqs. (7.3.2) and (7.3.3), the amplitude becomes ST 2 (1; 2; . . . ; n)

= F0

dτd¯τ 4τ2

b(0)˜b(0)˜c(0)c(0)

n

¯ i Vi (wi , w ¯ i) dwi dw

i=1

.

(7.3.4)

T2

All vertex operators are now on an equal footing. The form (7.3.4) could also have been derived directly, without the intermediate step of ﬁxing a vertex operator, and is valid even without vertex operators,

ZT 2 =

F0

dτd¯τ b(0)˜b(0)˜c(0)c(0) 2 . T 4τ2

(7.3.5)

This vacuum amplitude is quite interesting and will be our main focus. Not only does it reveal the essential diﬀerence between the ultraviolet behaviors of string theory and ﬁeld theory, but it has an important physical interpretation of its own. For 26 ﬂat dimensions, the matter and ghost path integrals (7.3.5) were evaluated in the previous section, with the result

ZT 2 = iV26

F0

dτd¯τ (4π 2 α τ2 )−13 |η(τ)|−48 . 4τ2

(7.3.6)

The amplitude has the important property of modular invariance. The product τ2 |η(τ)|4 is invariant by eq. (7.2.44), and it is easy to check that dτd¯τ τ22

(7.3.7)

is modular-invariant as well. Note the exponent −48, from the 24 leftand right-moving oscillators. The contribution of the ghosts cancels two sets of oscillators, leaving only the contribution of the transverse modes. The amplitude with n tachyons is given by including the vertex operator expectation value (7.2.4) and integrating over positions. For a general CFT the path integral on the torus is expressed as a trace in eq. (7.2.28). As long as there are d ≥ 2 noncompact ﬂat dimensions the ghosts still cancel two sets of bosonic operators, and the vacuum amplitude is

ZT 2 = Vd

F0

= iVd

F0

dτd¯τ 4τ2

dd k ˜ exp(−πτ2 α k 2 ) q hi −1 q¯hi −1 d (2π) ⊥

(7.3.8a)

i∈H

dτd¯τ ˜ (4π 2 α τ2 )−d/2 q hi −1 q¯hi −1 . 4τ2 ⊥ i∈H

(7.3.8b)

218

7 One-loop amplitudes

Here H⊥ is the closed string Hilbert space excluding the ghosts, the µ = 0, 1 oscillators, and the noncompact momenta. In order to understand the physics of these amplitudes it is useful to compare with the corresponding quantity in ﬁeld theory, the sum over all particle paths with the topology of a circle. This is given by

∞ dl dd k ZS1 (m ) = Vd exp[−(k 2 + m2 )l/2] (2π)d 0 2l ∞ dl = iVd (7.3.9) (2πl)−d/2 exp(−m2 l/2) . 0 2l This can be derived by gauge-ﬁxing the point-particle path integral, which was given as exercise 5.2. The result is intuitive: l is the modulus for the circle, 12 (k 2 + m2 ) is the world-line Hamiltonian, and the 2l in the denominator removes the overcounting from translation and reversal of the world-line coordinate. Let us now take this point-particle result, which is for a particle of given m2 , and sum over the actual spectrum of the string. As shown in chapter 4, the physical spectrum is in one-to-one correspondence with H⊥ , where the mass is related to the transverse weights by 2 (7.3.10) m2 = (h + ˜h − 2) α and with the constraint h = ˜h. It is useful to write the latter constraint in integral form, π

dθ (7.3.11) δh,˜h = exp i(h − ˜h)θ . −π 2π In writing this we are assuming that the diﬀerence h − ˜h is an integer. This is true in 26 ﬂat dimensions, and as we have discussed in the previous section it is necessary in general for modular invariance. Then ∞ dl π dθ ZS1 (m2i ) = iVd (2πl)−d/2 2l 2π 0 −π ⊥ 2

i∈H

×

= iVd where θ +

il/α

R

exp −(hi + ˜hi − 2)l/α + i(hi − ˜hi )θ

i∈H⊥

dτd¯τ ˜ (4π 2 α τ2 )−d/2 q hi −1 q¯hi −1 , 4τ2 ⊥

(7.3.12)

i∈H

= 2πτ. Here the region of integration R is

1 . (7.3.13) 2 Now let us interpret these results. The amplitude (7.3.9) for a single point particle diverges as l → 0. This is the usual UV divergence of quantum R:

τ2 > 0 ,

|τ1 | <

219

7.3 The torus amplitude

ﬁeld theory (note that l has units of spacetime length-squared). Summing this over the string spectrum as in eq. (7.3.12) only makes things worse, as all the states of the string contribute with the same sign. However, compare the sum (7.3.12) to the actual string amplitude (7.3.8). There is great similarity but also one remarkable diﬀerence. The integrands are identical but the regions of integration are diﬀerent: in the string it is the fundamental region 1 (7.3.14) , τ2 > 0 2 and in ﬁeld theory the larger region R. In the string amplitude, the UV divergent region is simply absent. We can also see this from the point of view of the momentum integral (7.3.8a). Since τ2 is bounded below, this integral (after contour rotation) is a convergent Gaussian. Another possible divergence comes from the limit τ2 → ∞, where the torus becomes very long. In this region, the string amplitude in 26 ﬂat dimensions has the expansion F0 :

iV26

∞ dτ2

2τ2

|τ| > 1 ,

|τ1 | <

(4π 2 α τ2 )−13 exp(4πτ2 ) + 242 + . . . ,

(7.3.15)

the asymptotic behavior being controlled by the lightest string states. The ﬁrst term in the series diverges, and the interpretation is clear from the ﬁeld theory point of view. The series is in order of increasing masssquared. The ﬁrst term is from the tachyon and diverges due to the positive exponential in the path sum (7.3.9). This divergence is therefore an artifact of a theory with a tachyon, and will not aﬄict more realistic string theories. Incidentally, this path sum can be deﬁned by analytic continuation from positive mass-squared, but there is still a pathology — the continued energy density is complex. This signiﬁes an instability, the tachyon ﬁeld rolling down its inverted potential. With a general CFT, eq. (7.3.12) becomes iVd

∞ dτ2

2τ2

(4π 2 α τ2 )−d/2

exp(−πα m2i τ2 ).

(7.3.16)

i

Because the tachyon is always present in bosonic string theory this again diverges; in the absence of tachyons it will converge. The torus illustrates a general principle, which holds for all string amplitudes: there is no UV region of moduli space that might give rise to high energy divergences. All limits are controlled by the lightest states, the long-distance physics. For the torus with vertex operators the τ-integration is still cut oﬀ as above, but there are more limits to consider as vertex operators approach one another. The same general principle applies to these as well, as we will develop further in chapter 9.

220

7 One-loop amplitudes

One might try to remove the UV divergence from ﬁeld theory by cutting oﬀ the l integral. Similarly, one might try to make an analogous modiﬁcation of string theory, for example replacing the usual fundamental region F0 with the region |τ1 | ≤ 12 , τ2 > 1 whose lower edge is straight. However, in either case this would spoil the consistency of the theory: unphysical negative norm states would no longer decouple. We have seen in the general discussion in section 5.4 that the coupling of a BRSTnull state is proportional to a total derivative on moduli space. For the fundamental region F0 , the apparent boundaries I and I and II and II respectively (shown in ﬁg. 5.2) are identiﬁed and the surface terms cancel. Modifying the region of integration introduces real boundaries. Surface terms on the modiﬁed moduli space will no longer cancel, so there is nonzero amplitude for null states and an inconsistent quantum theory. This is directly analogous to what happens in gravity if we try to make the theory ﬁnite by some brute force cutoﬀ on the short-distance physics: it is exceedingly diﬃcult to do this without spoiling the local spacetime symmetry and making the theory inconsistent. String theory manages in a subtle way to soften the short-distance behavior and eliminate the divergences without loss of spacetime gauge invariance. Physics of the vacuum amplitude Besides serving as a simple illustration of the behavior of string amplitudes, the one-loop vacuum amplitude has an interesting physical interpretation. In point-particle theory, ‘vacuum’ paths consist of any number n of disconnected circles. Including a factor of 1/n! for permutation symmetry and summing in n leads to

Z vac (m2 ) = exp ZS1 (m2 ) .

(7.3.17)

Translating into canonical ﬁeld theory, Z vac (m2 ) = 0| exp(−iHT )|0

= exp(−iρ0 Vd ) , where ρ0 is the vacuum energy density: i ZS (m2 ) . ρ0 = Vd 1

(7.3.18)

(7.3.19)

The l integral in ZS1 (m2 ) diverges as l → 0. In renormalizable ﬁeld theory this divergence is canceled by counterterms or supersymmetry. To get a rough insight into the physics we will cut oﬀ the integral at l ≥ and then take → 0, but drop divergent terms. With this prescription, ∞ dl 0

1 exp[−(k 2 + m2 )l/2] → − ln(k 2 + m2 ) 2l 2

(7.3.20)

221

7.3 The torus amplitude and i

∞ dl ∞ dk 0 0

2l

−∞

2π

exp[−(k 2 + m2 )l/2] →

ωk , 2

(7.3.21)

where ωk2 = k · k + m2 . Using the latter form, the vacuum energy density becomes

ρ0 =

dd−1 k ωk , (2π)d−1 2

(7.3.22)

which is just the sum of the zero-point energies of the modes of the ﬁeld. This is in spacetime, of course — we encountered the same kind of sum but on the world-sheet in section 1.3. The description of a quantum ﬁeld theory as a sum over particle paths is not as familiar as the description in terms of a sum over ﬁeld histories, but it is equivalent. In particular, the free path integral in ﬁeld theory can be found in most modern ﬁeld theory textbooks: 1 ln Z vac (m2 ) = − Tr ln(−∂2 + m2 ) 2 Vd dd k =− ln(k 2 + m2 ). 2 (2π)d

(7.3.23)

Using eq. (7.3.20), this is the same as the path sum result (7.3.9). In older quantum ﬁeld theory texts, one often reads that vacuum amplitudes such as eq. (7.3.18) are irrelevant because they give an overall phase that divides out of any expectation value. This is true if one is considering scattering experiments in a ﬁxed background and ignoring gravity, but there are at least two circumstances in which the vacuum energy density is quite important. The ﬁrst is in comparing the energy densities of diﬀerent states to determine which is the true ground state of a theory. It is very likely, for example, that the breaking of the electroweak S U(2)×U(1) symmetry is determined in part by such quantum corrections to the vacuum energy density. This was the original motivation that led to the Coleman–Weinberg formula (7.3.23). The generalization to a theory with particles of arbitrary spin is ρ0 =

i (−1)Fi ZS1 (m2i ). Vd i

(7.3.24)

The sum runs over all physical particle states. Each polarization counts separately, giving a factor of 2si + 1 for a particle of spin si in four dimensions. Here Fi is the spacetime fermion number, deﬁned mod 2, so fermions contribute with the opposite sign. The second circumstance is when one considers the coupling to gravity. The vacuum energy gives a source term in Einstein’s equation, the

222

7 One-loop amplitudes

cosmological constant, and so has observable eﬀects. In fact, this cosmological constant is a great challenge, because the fact that spacetime is approximately ﬂat and static means that its value is very small, −44 |ρ0 | < GeV 4 . ∼ 10

(7.3.25)

If one considers only the contributions to the vacuum energy from the vacuum ﬂuctuations of the known particles up to the currently explored energy (roughly the electroweak scale mew ), the zero-point energy is already of order m4ew ≈ 108 GeV 4 ,

(7.3.26)

52 orders of magnitude too large. The Higgs ﬁeld potential energy and the QCD vacuum energy are also far too large. Finding a mechanism that would account for the cancellation of the net cosmological constant to great accuracy has proven very diﬃcult. For example, in a supersymmetric theory, the contributions of degenerate fermions and bosons in the sum (7.3.24) cancel. But supersymmetry is not seen in nature, so it must be a broken symmetry. The cancellation is then imperfect, leaving again a remainder of at least m4ew . The cosmological constant problem is one of the most nagging diﬃculties, and therefore probably one of the best clues, in trying to ﬁnd a uniﬁed theory with gravity. What about the cosmological constant problem in string theory? At string tree level we had a consistent theory with a ﬂat metric, so the cosmological constant was zero. In fact we arranged this by hand, when we took 26 dimensions. One sees from the spacetime action (3.7.20) that there would otherwise be a tree-level potential energy proportional to D − 26. The one-loop vacuum energy density in bosonic string theory is nonzero, and is necessarily of the order of the string scale (ignoring the tachyon divergence). In four dimensions this would correspond to 1072 GeV 4 , which is again far too large. In a supersymmetric string theory, there will be a certain amount of cancellation, but again one expects a remainder of at least m4ew in a realistic theory with supersymmetry breaking. The cosmological constant problem is telling us that there is something that we still do not understand about the vacuum, in ﬁeld theory and string theory equally. We will return to this problem at various points. 7.4

Open and unoriented one-loop graphs The cylinder

The results for the torus are readily extended to the other surfaces of Euler number zero. For example, the vacuum amplitude from the cylinder

7.4 Open and unoriented one-loop graphs

223

σ2 σ1 (a)

(b)

(c) Fig. 7.1. (a) Cylinder in the limit of small t. (b) The amplitude separated into disk tadpole amplitudes and a closed string propagator. (c) Analogous ﬁeld theory graph. The heavy circles represent the tadpoles.

in the oriented theory is ZC2 =

∞ dt 0

= iVd

2t

Tro [exp(−2πtL0 )]

∞ dt 0

2t

(8π 2 α t)−d/2

∞ dt

exp[−2πt(hi − 1)]

i∈H⊥ o

(7.4.1) (8π 2 α t)−13 η(it)−24 . 2t 0 This can be obtained either by working out the path integral measure in terms of the ghost zero modes, as we did for the torus, or by guessing (correctly) that we should again sum the point-particle result (7.3.9) over the open string spectrum. In the ﬁrst line, the trace is over the full open string CFT, except that for omission of the ghost zero modes as denoted by the prime. In the ﬁnal line the trace has been carried out for the case of 26 ﬂat dimensions, with n Chan–Paton degrees of freedom. Including tachyon vertex operators is straightforward. The t → ∞ limit of the cylinder is much like the τ2 → ∞ limit of the torus. The cylinder looks like a long strip, and the leading asymptotics are given by the lightest open string states. As in the closed string there is a divergence, but only from the open string tachyon. The t → 0 limit is rather interesting. Unlike the case of the torus, there is no modular group acting to cut oﬀ the range of integration, and so the UV divergence of ﬁeld theory still seems to be present. However, we will see that, as with all divergences in string theory, this should actually be interpreted as a long-distance eﬀect. In the t → 0 limit the cylinder is very long as shown in ﬁgure 7.1(a). That is, it looks like a closed string appearing from the vacuum, propagating a distance, and then disappearing again into the vacuum. To make this clear, use the → iV26 n

2

224

7 One-loop amplitudes

modular transformation (7.2.44) η(it) = t−1/2 η(i/t)

(7.4.2)

and change variables to s = π/t, with the result ZC2 = i

V26 n2 2π(8π 2 α )13

∞

ds η(is/π)−24 .

(7.4.3)

0

Scaling down the metric by 1/t so that the cylinder has the usual closed string circumference 2π, the length of the cylinder is s. Expanding η(is/π)−24 = exp(2s)

∞

[1 − exp(−2ns)]−24

n=1

= exp(2s) + 24 + O(exp(−2s)) ,

(7.4.4)

one sees the expected asymptotics from expanding in a complete set of closed string states. In other words, if we think of σ 2 as world-sheet time and σ 1 as world-sheet space, ﬁgure 7.1(a) is a very short open string loop. If we reverse the roles of world-sheet space and time, it is a very long closed string world-line beginning and ending on boundary loops. In the Euclidean path integral, either description can be used, and each is useful in a diﬀerent limit of moduli space. The leading divergence in the vacuum amplitude is from the closed string tachyon. This is uninteresting and can be deﬁned by analytic continuation, ∞ 0

ds exp(βs) ≡ −

1 . β

(7.4.5)

The second term, from the massless closed string states, gives a divergence of the form 1/0 even with the continuation. To see the origin of this divergence, imagine separating the process as shown in ﬁgure 7.1(b). As we have discussed in section 6.6, there is a nonzero amplitude with one closed string vertex operator on the disk. This tadpole corresponds to a closed string appearing from or disappearing into the vacuum. In between the two tadpoles is the closed string propagator. In momentum space the massless propagator is proportional to 1/k 2 . Here, momentum conservation requires that the closed string appear from the vacuum with zero momentum, so the propagator diverges. This same kind of divergence occurs in quantum ﬁeld theory. Consider a massless scalar ﬁeld φ with a term linear in φ in the Lagrangian. There is then a vertex which connects to just a single propagator, and the graph shown in ﬁgure 7.1(c) exists. This diverges because the intermediate propagator is

1 . k 2 kµ =0

(7.4.6)

7.4 Open and unoriented one-loop graphs

225

This is a long-distance (IR) divergence, because poles in propagators are associated with propagation over long spacetime distances. UV and IR divergences in quantum ﬁeld theory have very diﬀerent physical origins. UV divergences usually signify a breakdown of the theory, the need for new physics at some short distance. IR divergences generally mean that we have asked the wrong question, or expanded in the wrong way. So it is here. In normal perturbation theory we expand around φ(x) = 0, or some other constant conﬁguration. For the action 1 − 2 g

d

d x

1 ∂µ φ∂µ φ + gΛφ 2

,

(7.4.7)

the equation of motion ∂2 φ = gΛ

(7.4.8)

does not allow φ(x) = 0 as a solution. Instead we must expand around a solution to the equation (7.4.8); any solution is necessarily positiondependent. The resulting amplitudes are free of divergences. This is true even though the right-hand side is a perturbation (we have included the factor of g appropriate to the disk), because it is a singular perturbation. In particular, the corrected background breaks some of the Poincar´e symmetry of the zeroth order solution. The situation is just the same in string theory. The disk tadpole is a source −Λ

˜

d26 x (−G)1/2 e−Φ

(7.4.9)

for both the dilaton and metric. Expanding around a solution to the corrected ﬁeld equations (with the metric and dilaton no longer constant) leads to ﬁnite amplitudes. The details are a bit intricate, and will be discussed somewhat further in chapter 9. Incidentally, in supersymmetric string theories if the tree-level background is invariant under supersymmetry, it usually receives no loop corrections. The pole (7.4.6) is the same kind of divergence encountered in the tree-level amplitudes, a resonance corresponding to propagation over long spacetime distances. If we add open string vertex operators to each end of the cylinder so that it represents a one-loop open string scattering amplitude, then the momentum k µ ﬂowing from one boundary to the other is in general nonzero. The large-s limit (7.4.4) then includes a factor of exp(−α k 2 s/2)

(7.4.10)

and the divergence becomes a momentum pole representing scattering of open strings into an intermediate closed string. Thus, as claimed in chapter 3, an open string theory must include closed strings as well. It is

226

7 One-loop amplitudes

curious that the mechanism that removes the UV divergence is diﬀerent in the closed and open string cases. In the closed string it is an eﬀective cutoﬀ on the modular integration; in the open string it is a reinterpretation of the dangerous limit of moduli space in terms of long distances. In eq. (7.4.1) the path integral on the cylinder has been related to a trace over the open string spectrum by cutting open the path integral with σ 2 treated as time. It can also be obtained in terms of the closed string Hilbert space by treating σ 1 as time. Let σ 2 be periodic with period 2π and the boundaries be at σ 1 = 0, s. The closed string appears in some state |B at σ 1 = 0 and then disappears in the same way at σ 1 = s. Including the measure insertions, the path integral is then proportional to ˜ 0 )]|B .

B|c0 b0 exp[−s(L0 + L (7.4.11) The boundary state |B is determined by the condition that ∂1 X µ , c1 , and b12 vanish on the boundary. In the Hamiltonian form these must annihilate |B , which in terms of the Laurent coeﬃcients is (αµ + ˜αµ−n )|B = (cn + ˜c−n )|B = (bn − ˜b−n )|B = 0 , all n . (7.4.12) n

This determines

|B ∝ (c0 + ˜c0 ) exp −

∞

−1

(n α−n · ˜α−n + b−n˜c−n + ˜b−n c−n ) |0; 0 . (7.4.13)

n=1

Using this in (7.4.11) gives the result (7.4.3), except that the normalization of |B is undetermined. This representation is useful in analyzing the t → 0 limit and the closed string poles. By comparing the string and ﬁeld theory calculations we can determine the disk tadpole Λ, but it will be more convenient to do this in the next chapter as a special case of a more general result. The Klein bottle The vacuum amplitude from the Klein bottle is ZK 2 =

∞ dt 0

= iVd

4t

#

˜ 0 )] Trc Ω exp[−2πt(L0 + L

∞ dt 0

4t

(4π 2 α t)−d/2

$

Ωi exp[−2πt(hi + ˜hi − 2)] ,

(7.4.14)

i∈H⊥ c

where the notation follows the cylinder (7.4.1). Relative to the torus in the oriented theory there is an extra factor of 12 from the projection operator 1 1 2 (1 + Ω). For the same reason there is an extra 2 in both the torus and cylinder amplitudes in the unoriented theory. One can also think of this ¯ To evaluate the trace as coming from the extra gauge invariance w → w. for 26 ﬂat dimensions, note that the only diagonal elements of Ω are those

7.4 Open and unoriented one-loop graphs

227

σ2 σ1 (a)

(b)

Fig. 7.2. (a) Klein bottle in the limit of small t as a cylinder capped by crosscaps. (b) The amplitude separated into RP2 tadpole amplitudes and a closed string propagator.

for which the right- and left-movers are in the same state, and these states contribute with Ω = +1. The trace is then eﬀectively over only one side, which is the same as the open string spectrum, except that the weight is doubled because the right- and left-movers make equal contributions. The result is ∞ dt (4π 2 α t)−13 η(2it)−24 . (7.4.15) ZK2 → iV26 4t 0 The modulus t runs over the same range 0 < t < ∞ as for the cylinder, so the t → 0 divergences are the same. Again the 1/0 pole has a longdistance interpretation in terms of a closed string pole. To see this consider the regions 0 ≤ σ 1 ≤ 2π , 0 ≤ σ 2 ≤ 2πt , 0 ≤ σ 1 ≤ π , 0 ≤ σ 2 ≤ 4πt , each of which is a fundamental region for the identiﬁcation ¯ + 2πit . w∼ = w + 2π ∼ = −w

(7.4.16a) (7.4.16b) (7.4.17)

In (7.4.16a), the left- and right-hand edges are periodically identiﬁed while the upper and lower edges are identiﬁed after a parity-reversal, giving an interpretation as a closed string loop weighted by Ω. For (7.4.16b), note that the identiﬁcations (7.4.17) imply also that ¯ + π) + 2πit . w∼ (7.4.18) = w + 4πit , w + π ∼ = −(w It follows that the upper and lower edges of the region (7.4.16b) are periodically identiﬁed. The left-hand edge is identiﬁed with itself after translation by half its length and reﬂection through the edge, and similarly the right-hand edge: this is the deﬁnition of a cross-cap. Thus we have the representation in ﬁgure 7.2(a) as a cylinder capped by two cross-caps. Rescaling by 1/2t, the cylinder has circumference 2π and length s = π/2t. After a modular transformation, the amplitude becomes ZK2

226 V26 =i 4π(8π 2 α )13

∞ 0

ds η(is/π)−24 .

(7.4.19)

228

7 One-loop amplitudes

σ2 σ1 (a)

(b)

Fig. 7.3. (a) M¨ obius in the limit of small t as a cylinder capped by one cross-cap. (b) The amplitude separated into D2 and RP2 tadpole amplitudes and a closed string propagator.

All the discussion of the cylinder divergence now applies, except that the tadpole is from the projective plane rather than the disk. The M¨obius strip For the M¨ obius strip, ZM2 = iVd

∞ dt 0

4t

(8π 2 α t)−d/2

Ωi exp[−2πt(hi − 1)]

(7.4.20)

i∈H⊥ o

This diﬀers from the result on the cylinder only by the Ω and the factor of 12 from the projection operator. In 26 ﬂat dimensions, the eﬀect of the operator Ω in the trace is an extra −1 at the even mass levels plus an appropriate accounting of the Chan–Paton factors. The oscillator trace is thus exp(2πt)

∞

[1 − (−1)n exp(−2πnt)]−24 = ϑ00 (0, 2it)−12 η(2it)−12 . (7.4.21)

n=1

For the S O(n) theory the 12 n(n + 1) symmetric states have Ω = +1 while the 12 n(n − 1) antisymmetric states have Ω = −1 for a net contribution of n. For the S p(k) theory these degeneracies are reversed, giving −n (recall that in our notation S p(k) corresponds to n = 2k Chan–Paton states). The amplitude is then ZM2 = ±inV26

∞ dt 0

4t

(8π 2 α t)−13 ϑ00 (0, 2it)−12 η(2it)−12 .

(7.4.22)

By the same construction as for the Klein bottle, the M¨ obius strip can be represented as a cylinder with a boundary at one end and a cross-cap on the other, as in ﬁgure 7.3(a). The length of the cylinder is now s = π/4t. By a modular transformation, the amplitude becomes ZM 2

213 V26 = ±2in 4π(8π 2 α )13

∞ 0

ds ϑ00 (0, 2is/π)−12 η(2is/π)−12 .

(7.4.23)

Exercises

229

As for the annulus this can be written as an operator expression, one of the boundary states in eq. (7.4.11) being replaced by an analogous cross-cap state |C . There is again a t → 0 divergence; it corresponds to the process of ﬁgure 7.3(a) with one tadpole from the disk and one from the projective plane. In the unoriented theory the divergences on these three surfaces combine into ∞ 24V26 13 2 (2 ∓ n) ds . (7.4.24) i 4π(8π 2 α )13 0 That is, the total tadpole is proportional to 213 ∓ n. For the gauge group S O(213 ) = S O(8192) this vanishes, the tadpoles from the disk and projective plane canceling. For the bosonic string this probably has no special signiﬁcance, but the analogous cancellation for S O(32) in superstring theory does. Exercises 7.1 Fill in the steps leading to the expectation value (7.2.4) of exponential operators on the torus. Show that it has the correct transformation under τ → −1/τ. 7.2 Derive the torus vacuum amplitude (7.3.6) by regulating and evaluating the determinants, as is done for the harmonic operator in appendix A. Show that a modular transformation just permutes the eigenvalues. [Compare exercise A.3.] (Reference: Polchinski (1986).) 7.3 Derive the Green’s function (7.2.3) by carrying out the eigenfunction sum (6.2.7). 7.4 Evaluate ∂w X µ (w)∂w Xµ (0) on the torus by representing it as a trace. Show that the result agrees with eq. (7.2.16). 7.5 (a) Obtain the leading behavior of the theta functions (7.2.37) as Im τ → ∞. (b) By using the modular transformations, do the same as τ → 0 along the imaginary axis. (c) A more exotic question: what if τ approaches a nonzero point on the real axis, along a path parallel to the imaginary axis? [Hint: the answer depends crucially on whether the point is rational or irrational.] 7.6 (a) Verify eqs. (7.3.20) and (7.3.21). (b) Evaluate ZS1 (m2 ) with a cutoﬀ on the l-integral in any dimension d. Show that the counterterms are analytic in m2 . Give the nonanalytic part for d = 4. (Reference: Coleman & Weinberg (1973).) 7.7 Obtain the amplitude for n open string tachyons on the cylinder

230

7 One-loop amplitudes

in a form analogous to the torus amplitude (eq. (7.3.6) combined with eq. (7.2.4)). The necessary Green’s function can be obtained from that on the torus by the method of images, using the representation of the cylinder as an involution of the torus. 7.8 Consider the amplitude from the previous problem, in the case that there are some vertex operators on each of the two boundaries. Show from the explicit result of the previous problem that the t → 0 limit gives a series of closed string poles as a function of the momentum ﬂowing from one boundary to the other. 7.9 If you have done both exercises 6.12 and 7.8, argue that unitarity relates the square of the former to the tachyon pole of the latter. Use this to ﬁnd the numerical relation between gc and go2 . We will do something similar to this, in a more roundabout way, in the next chapter. 7.10 Argue that unitarity of the cylinder amplitude with respect to intermediate open strings requires the Chan–Paton factors to satisfy Tr (λa λb )Tr (λc λd ) = Tr (λa λb λe λc λd λe ). Show that the gauge group U(n1 ) × U(n2 ) × . . . × U(nk ) is consistent with unitarity at tree level, but that this additional condition singles out U(n). 7.11 This exercise reproduces the ﬁrst appearance of D = 26 in string theory (Lovelace, 1971). Calculate the cylinder vacuum amplitude from the Coleman–Weinberg formula in the form (7.3.9), assuming D spacetime dimensions and D net sets of oscillators. Fix the constant in the open string mass-squared so that the tachyon remains at m2 = −1/α , as required by conformal invariance of the vertex operator. Insert an additional factor of exp(−α k 2 s/2) as in eq. (7.4.10) to simulate a scattering amplitude. Show that unless D = 26 and D = 24, the amplitude has branch cuts (rather than poles) in k 2 . 7.12 Repeat the previous exercise for the torus, showing in this case that the vacuum amplitude is modular-invariant only for D = 26 and D = 24 (Shapiro, 1972). 7.13 Carry out the steps, parallel to those used to derive the torus amplitude (7.3.6), to obtain the vacuum Klein bottle amplitude (7.4.15) from the string path integral. 7.14 Show that the amplitude (7.4.11) with the boundary state (7.4.13) reproduces the cylinder amplitude (7.4.3). 7.15 (a) Find the state |C corresponding to a cross-cap. (b) By formulae analogous to eq. (7.4.11), obtain the Klein bottle and M¨ obius strip vacuum amplitudes.

8 Toroidal compactiﬁcation and T -duality

Realistic compactiﬁcations of string theory will be the subject of the later chapters of volume two, but it is interesting to look now at the simplest compactiﬁcation of string theory, in which one or more dimensions are periodically identiﬁed. We ﬁrst consider the same compactiﬁcation in ﬁeld theory, encountering in particular the Kaluza–Klein uniﬁcation of gauge interactions and gravity. We then extend this to string theory, where several new and intrinsically stringy phenomena arise: winding states, enhanced gauge symmetries, T -duality, and D-branes. We are also led to consider slightly more complicated compactiﬁcations, namely orbifolds and orientifolds. 8.1

Toroidal compactiﬁcation in ﬁeld theory

In general relativity, the geometry of spacetime is dynamical. The three spatial dimensions we see are expanding and were once highly curved. It is a logical possibility that there are additional dimensions that remain small. In fact, this was put forward as early as 1914 as a means of unifying the electromagnetic and gravitational ﬁelds as components of a single higherdimensional ﬁeld. Consider the case of a ﬁve-dimensional theory, with x4 periodic, (8.1.1) x4 ∼ = x4 + 2πR , and with xµ noncompact for µ = 0, . . . , 3. This is toroidal compactiﬁcation. The ﬁve-dimensional metric separates into Gµν , Gµ4 , and G44 . From the four-dimensional point of view, these are a metric, a vector, and a scalar. Let us see this in detail, taking the more general case of D = d + 1 spacetime dimensions with xd periodic. Since we are still in ﬁeld theory we leave D arbitrary. Parameterize the metric as M N µ ν d µ 2 ds2 = GD MN dx dx = Gµν dx dx + Gdd (dx + Aµ dx ) .

231

(8.1.2)

232

8 Toroidal compactiﬁcation and T -duality

Indices M, N run over all dimensions 0, . . . , d and indices µ, ν run only over noncompact dimensions 0, . . . , d − 1. We designate the full D-dimensional D metric by GD MN . One should note that Gµν = Gµν ; in d-dimensional actions we raise and lower indices with Gµν . For now the ﬁelds Gµν , Gdd , and Aµ are allowed to depend only on the noncompact coordinates xµ . The form (8.1.2) is the most general metric invariant under translations of xd . This form still allows d-dimensional reparameterizations xµ (xν ) and also reparameterizations xd = xd + λ(xµ ) .

(8.1.3)

Aµ = Aµ − ∂µ λ ,

(8.1.4)

Under the latter,

so gauge transformations arise as part of the higher-dimensional coordinate group. This is the Kaluza–Klein mechanism. To see the eﬀect of xd -dependence, consider a massless scalar φ in D dimensions, where for simplicity the metric in the compact direction is Gdd = 1. The momentum in the periodic dimension is quantized, pd = n/R. Expand the xd -dependence of φ in a complete set, φ(xM ) =

∞

φn (xµ ) exp(inxd /R) .

(8.1.5)

n=−∞

The D-dimensional wave equation ∂M ∂M φ = 0 becomes n2 φn (xµ ) . (8.1.6) R2 The modes φn of the D-dimensional ﬁeld thus become an inﬁnite tower of d-dimensional ﬁelds, labeled by n. The d-dimensional mass-squared ∂µ ∂µ φn (xµ ) =

n2 (8.1.7) R2 is nonzero for all ﬁelds with nonvanishing pd . At energies small compared to R −1 , only the xd -independent ﬁelds remain and the physics is d-dimensional. At energies above R −1 , one sees the tower of Kaluza–Klein states. The charge corresponding to the Kaluza–Klein gauge invariance (8.1.3) is the pd -momentum. In this simple example, all ﬁelds carrying the Kaluza– Klein charge are massive. More generally, with higher spin ﬁelds and curved backgrounds, there can be massless charged ﬁelds. The eﬀective action for the massless ﬁelds is always an important object to consider. Deﬁne Gdd = e2σ . The Ricci scalar for the metric (8.1.2) is − p µ pµ =

1 R = Rd − 2e−σ ∇2 eσ − e2σ Fµν F µν , 4

(8.1.8)

8.1 Toroidal compactiﬁcation in ﬁeld theory

233

where R is constructed from GD MN and Rd from Gµν . The graviton-dilaton action (3.7.20) becomes

1 dD x (−GD )1/2 e−2Φ (R + 4∇µ Φ∇µ Φ) S1 = 2κ20 πR = 2 dd x (−Gd )1/2 e−2Φ+σ κ0 1 2σ µ µ µν × Rd − 4∂µ Φ∂ σ + 4∂µ Φ∂ Φ − e Fµν F 4 πR = 2 dd x (−Gd )1/2 e−2Φd κ0 1 × Rd − ∂µ σ∂µ σ + 4∂µ Φd ∂µ Φd − e2σ Fµν F µν , 4

(8.1.9)

giving kinetic terms for all the massless ﬁelds. Here Gd denotes the determinant of Gµν , and in the last line we have introduced the eﬀective d-dimensional dilaton, Φd = Φ − σ/2. The apparent wrong sign of the dilaton kinetic term is illusory because mixing of the graviton with the trace of the metric must also be taken into account. This is most easily done by means of a Weyl transformation as in eq. (3.7.25), and the resulting kinetic term has the correct sign. The ﬁeld equations do not determine the radius of the compact dimension:1 a ﬂat metric and constant dilaton are a solution for any values of Φ and σ. In other words, there is no potential energy for Φ or for σ, and so these ﬁelds are massless, much like Goldstone bosons. The diﬀerent values of Φ and σ label degenerate conﬁgurations (or states in the quantum theory), and a state in which these ﬁelds are slowly varying has energy only from the gradient. The diﬀerence from the Goldstone phenomenon is that the degenerate states are not related to one another by any symmetry, and in fact the physics depends on Φ and σ. In this bosonic theory the degeneracy is accidental and the one-loop energy, discussed in the previous chapter, breaks the degeneracy. In supersymmetric theories, the existence of physically inequivalent but degenerate vacua is quite common and plays an important role in understanding the dynamics. The massless ﬁelds, which label the inequivalent vacua, are called moduli. In nature, 1

Of course, only the invariant radius ρ = Reσ distinguishes physically inequivalent solutions. We could set R to some convenient value, say α1/2 , or to unity if we use dimensionless variables, but it is convenient to leave it general; often it will be more convenient to set Gdd to unity instead. Similarly we could set κ20 to be the appropriate power of α by an additive shift of Φ, but again it does not hurt to leave it general and we will see in chapter 13 that it is natural to ﬁx the additive normalization of Φ in a diﬀerent way.

234

8 Toroidal compactiﬁcation and T -duality

supersymmetry breaking must almost surely give mass to all moduli, else they would mediate inﬁnite-range interactions of roughly gravitational strength. We will see in chapter 14 that in some string theories the dilaton itself is the radius of a hidden dimension, one that is not evident in string perturbation theory. ˜ µ , the covariant derivative is Deﬁning Aµ = R A ˜µ , ∂µ + ipd Aµ = ∂µ + inA

(8.1.10)

so that the charges are integers. The d-dimensional gauge and gravitational ˜ µν ˜µν F couplings are conventionally deﬁned as follows. The coeﬃcient of F 2 in the Lagrangian density is deﬁned to be −1/4gd , and that of Rd is deﬁned to be 1/2κ2d . The gauge coupling is thus determined in terms of the gravitational coupling, gd2 =

κ20 e2Φd 2κ2d = . πR 3 e2σ ρ2

(8.1.11)

The d-dimensional and D-dimensional gravitational couplings are related 2πρ 1 = 2 , 2 κ κd

(8.1.12)

2πρ being the volume of the compactiﬁed dimension. The antisymmetric tensor also gives rise to a gauge symmetry by a generalization of the Kaluza–Klein mechanism. Separating BMN into Bµν and Aµ = Bdµ , the gauge parameter ζM deﬁned in eq. (3.7.7) separates into a d-dimensional antisymmetric tensor transformation ζµ and an ordinary gauge invariance ζd . The gauge ﬁeld is Bdµ and the ﬁeld strength Hdµν . The antisymmetric tensor action becomes

1 dD x (−GD )1/2 e−2Φ HMNL H MNL 24κ20 πR d 1/2 −2Φd ˜ ˜ µνλ + 3e−2σ H H µν . =− d x (−G ) e H H d µνλ dµν d 12κ20 (8.1.13)

S2 = −

We have deﬁned ˜ µνλ = (∂µ Bνλ − Aµ Hdνλ ) + cyclic permutations . H

(8.1.14)

The term proportional to the vector potential arises from the inverse metric GMN . It is known as a Chern–Simons term, this signifying the antisymmetrized combination of one gauge potential and any number of ﬁeld strengths. Such terms appear in many places in supersymmetric theories, and are associated with interesting physical eﬀects. Notice in

8.2 Toroidal compactiﬁcation in CFT

235

˜ µνλ is gauge-invariant because the variation (8.1.4) of Aµ particular that H is canceled by a variation Bνλ = Bνλ − λHdνλ .

(8.1.15)

There is no way to couple the potential BMN minimally to other ﬁelds, and so unlike the Kaluza–Klein case there are no ﬁelds charged under the antisymmetric tensor gauge symmetry; this will be diﬀerent in string theory.

8.2

Toroidal compactiﬁcation in CFT

Now we consider the conformal ﬁeld theory of a single periodic scalar ﬁeld, X∼ = X + 2πR .

(8.2.1)

To keep the equations uncluttered we drop the superscript from X d and world-sheet action is as in the noncompact theory, set 2 Gdd = 1. The , so the equations of motion, operator products, and ¯ d z ∂X ∂X/2πα energy-momentum tensor are unchanged. In particular, the theory is still conformally invariant. The periodicity has two eﬀects. First, string states must be single-valued under the identiﬁcation (8.2.1). That is, the operator exp(2πiRp) which translates strings once around the periodic dimension must leave states invariant, so the center-of-mass momentum is quantized k=

n , R

n∈Z.

(8.2.2)

This is just as in ﬁeld theory. The second eﬀect is special to string theory. A closed string may now wind around the compact direction, X(σ + 2π) = X(σ) + 2πRw ,

w∈Z.

(8.2.3)

The integer w is the winding number. States of winding numbers +1, 0, and −1 are shown in ﬁgure 8.1(a). From the point of view of the world-sheet ﬁeld theory, strings of nonzero winding number are topological solitons, states with a topologically nontrivial ﬁeld conﬁguration. A consistent string theory must include the winding number states: by the usual splitting–joining process a w = 0 string can turn into a w = +1, w = −1 pair as shown in ﬁgure 8.1(b). It is easy to see that winding number is always conserved as in this example. To describe the states of the closed string CFT, consider the Laurent

236

8 Toroidal compactiﬁcation and T -duality

X _1

0

+1

Y (a)

X 0

Y

_1

+1

(b)

Fig. 8.1. (a) Closed oriented strings of winding number w = +1, 0, −1. For illustration, one compact dimension X and one noncompact dimension Y are shown. (b) Transition of a w = 0 string into w = +1 and w = −1 strings.

expansions ∂X(z) = −i

1/2 ∞ α

αm

m+1 m=−∞ z

2

,

¯ z ) = −i α ∂X(¯ 2

1/2 ∞

˜αm . (8.2.4) m+1 ¯ z m=−∞

The total change in the coordinate X in going around the string is

2πRw =

¯ = 2π(α /2)1/2 (α0 − ˜α0 ) . (dz ∂X + d¯z ∂X)

(8.2.5)

The total Noether momentum is p=

1 2πα

¯ = (2α )−1/2 (α0 + ˜α0 ) . (dz ∂X − d¯z ∂X)

(8.2.6)

For a noncompact dimension this would give the usual relation α0 = ˜α0 = p(α /2)1/2 , but for the periodic dimension we have '

(1/2

n wR + , R α ' wR n (1/2 − . α˜0 = pR ≡ 2/α R α The Virasoro generators are pL ≡ 2/α

α0 =

(8.2.7a) (8.2.7b)

∞ α p2L L0 = α−n αn , + 4 n=1

(8.2.8a)

∞ 2 ˜ 0 = α pR + ˜α−n ˜αn . L 4 n=1

(8.2.8b)

237

8.2 Toroidal compactiﬁcation in CFT The partition function The partition function for X is now

˜

(q¯ q )−1/24 Tr q L0 q¯L0 = |η(τ)|−2 = |η(τ)|

∞

2

2

q α pL /4 q¯α pR /4

n,w=−∞ ∞ −2

exp −πτ2

n,w=−∞

α n2 w 2 R 2 + R2 α

+ 2πiτ1 nw

.

(8.2.9)

The oscillator sum is the same as in the noncompact case, while the momentum integration is replaced by a sum over n and w. Modular invariance is not manifest, but can be made so by using the Poisson resummation formula, ∞

2

exp(−πan + 2πibn) = a

n=−∞

∞

π(m − b)2 exp − . a m=−∞

−1/2

(8.2.10)

The partition function becomes 2πRZX (τ)

∞

exp −

m,w=−∞

πR 2 |m − wτ|2 α τ2

.

(8.2.11)

Here ZX (τ) is the modular-invariant expression (7.2.9) from the noncompact theory. The sum is obviously invariant under τ → τ + 1, by making a change of variables m → m + w. It is also invariant under τ → −1/τ with m → −w, w → m. The expression (8.2.11) has a simple path integral interpretation. Summing over all genus one world-sheets in a periodic spacetime, each nontrivial closed curve on the world-sheet can be wound around the compact direction, X(σ 1 + 2π, σ 2 ) = X(σ 1 , σ 2 ) + 2πwR , X(σ 1 + 2πτ1 , σ 2 + 2πτ2 ) = X(σ 1 , σ 2 ) + 2πmR .

(8.2.12a) (8.2.12b)

That is, the path integral breaks up into topologically distinct sectors labeled by w and m. The Gaussian path integration can be done by the usual strategy of writing X as a classical solution of the correct periodicity, Xcl = σ 1 wR + σ 2 (m − wτ1 )R/τ2 ,

(8.2.13)

plus a quantum piece with periodic boundary conditions. The action separates into a classical piece plus a quantum piece. The path integral over the quantum piece is just as in the noncompact case, while the classical action appears as the exponent in eq. (8.2.11). The eﬀect of the

238

8 Toroidal compactiﬁcation and T -duality

modular transformations on the periodicities is then simply to change the summation variables m and w. Vertex operators In order to create a winding state, which has α0 = ˜α0 , we need independent variables xL and xR , with [ xL , pL ] = [ xR , pR ] = i .

(8.2.14)

The ﬁeld X then splits into holomorphic and antiholomorphic parts, X(z, ¯z ) = XL (z) + XR (¯z ) , with

α α XL (z) = xL − i pL ln z + i 2 2

α α XR (¯z ) = xR − i pR ln ¯z + i 2 2

1/2 ∞ m=−∞ m=0

1/2 ∞

m=−∞ m=0

(8.2.15) αm , mz m

(8.2.16a)

˜αm . m¯z m

(8.2.16b)

If we restrict attention to states with kL = kR and to operators constructed only from the sum XL (z)+XR (¯z ), this reduces to the CFT for a noncompact dimension. Let us now discuss the OPE and vertex operators, ﬁrst in a slightly heuristic way and then ﬁlling in the details. It is easy to guess what the operator products should be, since the usual XX operator product −(α /2) ln(z12 ¯z12 ) separates into a sum of holomorphic plus antiholomorphic functions. Thus, α ln z12 , 2 XL (z1 )XR (¯z2 ) ∼ 0 .

XL (z1 )XL (z2 ) ∼ −

XR (¯z1 )XR (¯z2 ) ∼ −

α ln ¯z12 , 2

(8.2.17a) (8.2.17b)

The operator corresponding to the state |0; kL , kR would be VkL kR (z, ¯z ) = : eikL XL (z)+ikR XR (¯z ) :

(8.2.18)

with OPE α k kL /2 α kR kR /2 ¯z12 V(k+k )L (k+k )R (z2 , ¯z2 )

VkL kR (z1 , ¯z1 )VkL kR (z2 , ¯z2 ) ∼ z12 L

. (8.2.19) One encounters branch cuts in various expressions, which is not surprising because the ﬁeld X is no longer single-valued. It is important, however, that the full vertex operator OPE is single-valued: the net phase when z1 circles z2 is exp[πiα (kL kL − kR kR )] = exp[2πi(nw + wn )] = 1 .

(8.2.20)

This is as it must be in order for the string amplitudes to be well deﬁned.

239

8.2 Toroidal compactiﬁcation in CFT A technicality

The above is essentially correct, but we have been cavalier with the location of the branch cut in the logarithm. One can see a problem in the OPE (8.2.19) if one interchanges the points z1 and z2 and the momenta k and k . The left-hand side is symmetric, but the right-hand side changes by exp[πi(nw + wn )] and so changes sign if nw + wn is odd. One can also see the problem as follows. From the mode expansion one may derive the equal time (|z1 | = |z2 |) commutator [ XL (z1 ), XL (z2 ) ] =

πiα sign(σ11 − σ21 ) . 2

(8.2.21)

The CBH formula then shows that if we deﬁne the operators simply as creation–annihilation ordered, then for nw + wn odd the operators VkL kR and VkL kR anticommute rather than commute; the invisible branch cut (8.2.20) is separated into two visible ones. A correct oscillator expression for the vertex operator (there is some arbitrariness in the phase) is VkL kR (z, ¯z ) = exp[πi(kL − kR )(pL + pR )α /4] ◦◦ eikL XL (z)+ikR XR (¯z ) ◦◦ ,

(8.2.22)

where as usual the ps are momentum operators and the ks are numbers (the momenta carried by the given vertex operator). When VkL kR and VkL kR are commuted past each other, the additional factors, known as cocycles, give an additional phase #

exp πi (kL − kR )(kL + kR ) − (kL − kR )(kL + kR ) α /4

$

= exp[πi(nw − wn )] ,

(8.2.23)

removing the branch cuts. For most purposes one can ignore this complication and work with the simpler expressions of the previous paragraph; the cocycle aﬀects only the relative signs of certain amplitudes. The general X µ path integral (6.2.18) factorizes in an obvious way into holomorphic times antiholomorphic, so that one can simply replace n

|zij |α ki kj →

i,j=1 i<j

and also replace the 2πδ(

n α kLi kLj /2 α kRi kRj /2

zij

¯zij

,

(8.2.24)

i,j=1 i<j

k) from the noncompact integral over x0 by 2πRδΣi ni ,0 δΣi wi ,0 .

(8.2.25)

Again this is naive; the precise expression (8.2.22) gives some additional signs. The reader can readily calculate an amplitude with four vertex operators, which is similar to the Virasoro–Shapiro amplitude.

240

8 Toroidal compactiﬁcation and T -duality DDF operators

We have just seen that the exponential operator can be separated into holomorphic and antiholomorphic parts. This has many applications. We now use it to tie up a loose end from the end of chapter 4, the DDF operators. We will work with light-cone coordinates X ± = 2−1/2 (X 0 ± X 1 ), whose holomorphic parts have the OPE (we drop the subscript L) X + (z)X − (0) ∼

α ln z , 2

X + (z)X + (0) ∼ X − (z)X − (0) ∼ 0 .

(8.2.26)

It follows that the operators V i (nk0 , z) = ∂X i (z)eink0 X

+ (z)

(2/α )1/2

(8.2.27)

with i transverse are (1, 0) primary ﬁelds with OPE V i (nk0 , z) V j (mk0 , 0) ∼ − −

δ ij i(n+m)k0 X + (0) e z2

+ ink0 δ ij ∂X + (0)ei(n+m)k0 X (0) . z

(8.2.28)

Deﬁne the DDF operators by

Ain

=

dz i V (nk0 , z) . 2π

(8.2.29)

The residue of the z −1 term in the OPE (8.2.28) is a total derivative except when n + m = 0, and so the DDF operators satisfy an oscillator algebra [Aim , Ajn ] = mδ ij δm,−n

α k0 p+ , 2

(8.2.30)

which is an oscillator algebra, up to normalization. Consider the action of the DDF operators on a state of given momentum q. The OPE of V i (nk0 , z) with the vertex operator for this state − involves z −α nk0 q /2 and so is single-valued if we ﬁx k0 = 2/α q − . The contour integral (8.2.29) then deﬁnes a sensible operator in this sector. The DDF operators, being integrals of (1, 0) tensors, commute with the Virasoro generators and so take physical states into physical states. It follows that we can build physical states by acting with the DDF operators on the oscillator ground state. The states obtained in this way are in one-to-one correspondence with the light-cone states, and so are a complete set of physical states. The DDF operators do not involve X − and so these states have no α− −m excitations. They are therefore the same states given by the more general but less explicit construction (4.4.19).

8.3 Closed strings and T -duality 8.3

241

Closed strings and T -duality

Now that we are looking at string theory we take D = 26, with only X 25 ˜ 0 ) conditions are periodic at ﬁrst. The mass-shell (L0 and L 4 (N − 1) α 4 ˜ = (kR25 )2 + (N − 1) , α

m2 = −k µ kµ = (kL25 )2 +

(8.3.1)

or n2 w2R2 2 ˜ − 2) , + + (N + N 2 2 R α α ˜ . 0 = nw + N − N

m2 =

(8.3.2a) (8.3.2b)

One recognizes four contributions to the mass-squared: those from the compact momentum, the potential energy of the winding string, the oscillators, and the zero-point energy. The discussion of the physical spectrum in chapter 4 was valid for an arbitrary compactiﬁcation, so we get the correct counting of states by considering only the transverse oscillators M = 2, . . . , 25. Let us ﬁrst recover the ﬁeld theory results for the massless spectrum. At a generic value of R, the only way a state can be massless is if n = w = 0 ˜ = 1. These are the same 242 states as in the noncompact and N = N theory, but it is useful now to separate them according to whether the oscillations are in the spacetime directions µ or the internal direction 25: αν−1 |0; k , αµ−1 ˜

25 µ (αµ−1 ˜α25 α−1 )|0; k , −1 + α−1 ˜

25 µ α25 α−1 )|0; k , (αµ−1 ˜ −1 − α−1 ˜

α25 α25 −1 ˜ −1 |0; k .

(8.3.3)

The ﬁrst of these breaks up further into a 25-dimensional graviton plus dilaton plus antisymmetric tensor. The second, a graviton with one spacetime and one internal index, is the Kaluza–Klein vector. The third state is the vector from the antisymmetric tensor. The ﬁnal state is a scalar, the modulus for the radius of the compact direction. Its vertex operator, ¯ 25 eik·X :, is a perturbation of the metric G25,25 . This is the same : ∂X 25 ∂X massless spectrum as found by considering the low energy ﬁeld theory. It is interesting to consider the charges of massive states under the U(1) × U(1) symmetry. Again the Kaluza–Klein gauge symmetry is from X 25 -translations, so the corresponding charge is the compact momentum p25 . For the antisymmetric tensor gauge symmetry, consider the zeromomentum vertex operator, which measures the charge of any state to which it couples. It is proportional to ¯ 25 − ∂X 25 ∂X ¯ µ = ∂(X ¯ µ) . ¯ 25 ∂X µ ) − ∂(X 25 ∂X ∂X µ ∂X

(8.3.4)

242

8 Toroidal compactiﬁcation and T -duality

This is a total derivative, but it does not integrate to zero in a winding state because X 25 is not single-valued. So the Bµ,25 charge is the winding number. This is our ﬁrst of several examples of ‘stringy’ physics in this compactiﬁcation: in ﬁeld theory there were no states carrying this charge. Let us check the gauge couplings explicitly from the string three-point coupling. The gauge boson vertex operators are 21/2 gc,25 ¯ 25 ± ∂X 25 ∂X ¯ µ )eik·X : . : (∂X µ ∂X (8.3.5) α We have deﬁned the 25-dimensional string coupling gc,25 = gc (2πR)−1/2 . The factor (2πR)−1/2 is from normalizing the zero-mode wavefunction. The vertex operator for a tachyon with general compact momentum and winding is gc,25 : eikL ·XL (z)+ikR ·XR (¯z ) : .

(8.3.6)

The three-point amplitude for one gauge boson and two tachyons, with the tachyons having nonzero compact momenta and winding, is similar to eq. (6.6.14), −2−1/2 πigc,25 (2π)25 δ 25 ( → −23/2 πigc,25 (2π)25 δ 25 (

µ 25 25 i ki )k23 (kL23 ± kR23 ) µ 25 25 i ki )k2 (kL2 ± kR2 ) .

(8.3.7)

In the second line we have taken the gauge boson momentum k1 → 0, which deﬁnes the gauge coupling. The two gauge bosons thus couple 25 ±k 25 , the compact momentum and winding, as expected. respectively to kL2 R2 By considering also the amplitude with a graviton replacing the gauge boson, one can recover the same relations (8.1.11), (8.1.12) between the various couplings as found from the eﬀective action. Enhanced gauge symmetries Everything so far is almost exactly as it would be in ﬁeld theory with one compact dimension. The 26-dimensional graviton gives rise to a 25dimensional graviton plus vector plus modulus, and so on. The one stringy eﬀect is that winding states carry the Bµ,25 charge. Further and more striking eﬀects arise at special compactiﬁcation radii. Our discussion of the massless spectrum omitted states that are massless only for special values of the radius R. The richest case is R = α 1/2 , where 25 = (n ± w)α−1/2 and the condition for a massless state is kL,R ˜ = 4. (n + w)2 + 4N = (n − w)2 + 4N (8.3.8) ˜ = 1, there are now In addition to the generic solution n = w = 0, N = N also ˜ = 1 , n = −w = ±1, N = 1, N ˜ = 0 , (8.3.9) n = w = ±1, N = 0, N

8.3 Closed strings and T -duality

243

and ˜ =0, n = ±2, w = N = N

˜ =0. w = ±2, n = N = N

(8.3.10)

The states (8.3.9) include four new gauge bosons, with vertex operators ¯ µ eik·X exp[±2iα−1/2 X 25 ] : , : ∂X µ eik·X exp[±2iα−1/2 X 25 ] : . (8.3.11) : ∂X L R The precise deﬁnition of the exponential operators is as discussed in the previous section. These states have internal momentum and winding number, so they carry the Kaluza–Klein and antisymmetric tensor gauge charges. The only consistent theory of charged massless vectors is nonAbelian gauge theory, so the new gauge bosons must combine with the old into a non-Abelian theory. It is now useful to work with the basis ¯ µ and ∂X µ ∂X ¯ 25 for the earlier vectors. The ﬁrst of these couples to ∂X 25 ∂X 25 kL , under which the ﬁrst pair of states in (8.3.11) carries charges ±1 and the second pair is neutral. The second similarly couples to kR25 and so to the other pair. This identiﬁes the gauge group as S U(2) × S U(2), with the ¯ µ forming one S U(2) and the three involving three vectors involving ∂X µ ∂X forming the other. To exhibit the S U(2) × S U(2) deﬁne the three (1, 0) currents j 1 (z) = : cos[2α−1/2 XL25 (z)] : , 2

j (z) = : sin[2α 3

j (z) =

−1/2

i∂XL25 (z)/α

XL25 (z)] : 1/2 .

,

(8.3.12a) (8.3.12b) (8.3.12c)

These have been normalized to have the OPE δ ij ijk k +i (8.3.13) j (0) . 2 2z z There is of course also a similar OPE of (0, 1) currents ˜ i . The single pole term implies that the corresponding charges form an S U(2) algebra. In fact, since the currents are holomorphic there is an inﬁnite-dimensional algebra formed by the Laurent coeﬃcients j i (z)j j (0) ∼

i

j (z) =

∞

jmi

m=−∞

z m+1

,

(8.3.14a)

m k , (8.3.14b) δm,−n δ ij + iijk jm+n 2 known variously as a current algebra, aﬃne Lie algebra, or Kac–Moody algebra. We will encounter such algebras extensively beginning in chapter 11. It will be shown that the coeﬃcient of the z −2 term is quantized, with the value in (8.3.13) being the minimum, so this is known as a level one S U(2) current algebra. The existence of this S U(2) × S U(2) symmetry is the ﬁrst indication that string theory sees spacetime geometry diﬀerently from what we are used [ jmi , jnj ] =

244

8 Toroidal compactiﬁcation and T -duality

to in ﬁeld theory, where only a U(1) × U(1) symmetry is evident at any radius. The j 3 and j 1,2 currents appear to have very diﬀerent origins, as oscillator excitations and as winding–momentum excitations respectively, and their expressions in terms of X 25 are very diﬀerent, but in their action on the string spectrum they are related to one another by symmetry. The extra vectors are massless because the energies of internal momentum and winding are oﬀset by the negative zero-point energy. The reader might therefore suspect that these could arise only in theories with tachyons, but we will see otherwise in volume two. Scales and couplings At the S U(2) × S U(2) radius, the relation (8.1.11) becomes2 2 g25 = 2κ225 /α .

(8.3.15)

This relation continues to hold if we compactify more dimensions, each side scaling the same way, so in particular in four dimensions g42 = 2κ24 /α .

(8.3.16)

In nature, the non-Abelian gauge couplings g42 are within an order of magnitude of unity, so this implies that the string length α1/2 is not too far from the gravitational length κ4 . We can also think about this as follows. The four-dimensional gauge coupling is dimensionless, but the four-dimensional gravitational coupling is dimensional. Deﬁne an eﬀective dimensionless gravitational coupling depending on the energy scale E, 2 (E) = κ24 E 2 , gG,4

(8.3.17)

which increases quadratically with energy. Of course, the ‘dimensionless’ gauge couplings run with energy, but this is slow compared to the dimensional scaling of the gravitational coupling. Then by eq. (8.3.16), the string mass scale is where the gauge and gravitational couplings are roughly 2 (E) ≈ g 2 . equal, gG,4 4 It is also interesting to consider the possibility that in compactifying down to four dimensions, some of the compactiﬁed dimensions are larger than others. This can best be analyzed by starting at low energy and working upwards. At low energy there is a dimensionless gauge coupling g42 . At an energy ρ−1 5 , one compact dimension becomes visible and the physics 2

To be precise, this was the coupling of an (n, w) = (1, 0) state. The non-Abelian gauge bosons have (|n|, |w|) = (1, 1) and so their coupling, which is the conventional particle physics deﬁnition 2 = 4κ225 /α . We will see in chapter 18 that this holds for all level of the SU(2) coupling, is gSU(2),25 one current algebras.

8.3 Closed strings and T -duality

245

becomes ﬁve-dimensional (for simplicity we consider only one dimension at this scale; with more dimensions the conclusion is strengthened). The eﬀective ﬁve-dimensional coupling is g52 = 2πρ5 g42 .

(8.3.18)

This follows from the eﬀective action, as in eq. (8.1.12). The coupling g52 has units of length. That is, ﬁve-dimensional Yang–Mills theory, like four-dimensional gravity, is a nonrenormalizable theory. Correspondingly, the eﬀective dimensionless coupling gˆ 52 = g52 E = 2πρ5 Eg42

(8.3.19)

grows linearly with energy. However, g42 is not much less than one in nature, so the coupling quickly becomes strong at high energy; presumably string theory is then needed to make the short-distance behavior sensible. So the compactiﬁcation scale must be close to the string scale, which as we have already learned is close to the gravitational scale. This analysis is modiﬁed somewhat in open string theory, and when the recent understanding of strongly coupled string theory is taken into account; it is possible, but unlikely, that string or Kaluza–Klein excitations lie far below the Planck scale. Higgs mechanism It is instructive to consider what happens as we move R away from the S U(2) × S U(2) radius. The extra gauge bosons now acquire masses m=

2 |R 2 − α | 1/2 ≈ R − α , Rα α

(8.3.20)

the approximation being for radii close to the S U(2) × S U(2) radius α1/2 . Close to this radius the mass is much less than the string scale and we should be able to understand the physics in terms of low energy ﬁeld theory. There is only one way to give such a mass to a gauge boson, spontaneous symmetry breaking, and that is just what is happening here. At R = α 1/2 there are ten massless scalars: the dilaton, the modulus G25,25 , the four states in (8.3.9) where a 25-oscillator is excited, and the four states (8.3.10). The last nine are created by the vertex operators : j i (z)˜ j (¯z )eik·X(z,¯z ) : .

(8.3.21)

The index i is a vector under the left-moving S U(2) and the index j is a vector under the right-moving S U(2). That is, these transform as (3,3) under S U(2) × S U(2). In particular, the modulus for the radius is j 3˜ 3 . Moving away from the S U(2) × S U(2) radius gives an expectation value

246

8 Toroidal compactiﬁcation and T -duality

to this ﬁeld and so breaks the gauge symmetry down to the U(1) × U(1) that leaves the 3-axes invariant. Indeed, this is the gauge group at generic radii. Near the S U(2) radius, the mass (8.3.20) is linear in the magnitude |R − α1/2 | of the modulus, as usual in spontaneous breaking. Let us label the spacetime ﬁelds corresponding to the nine massless scalars as Mij . A change in the modulus is M33 = 0, but it is instructive to look for more general backgrounds near the S U(2) × S U(2) point. The interactions of these massless ﬁelds will be described by some spacetime action, which will include a potential U(M). These ﬁelds are massless at the symmetric point so there is no term of order M 2 in U, but an (S U(2) × S U(2))-invariant cubic term is possible,

U(M) ∝ ijk i j k Mii Mjj Mkk = det M .

(8.3.22)

It is not hard to verify from the string three-point amplitude that this term does appear. Being cubic in bosons it is unbounded below, but like the tachyon this is an irrelevant artifact of bosonic string theory. It is interesting to look for static classical solutions, ignoring the fact that they are unstable to small variations. In order to have a static background solution, the graviton and dilaton ﬁeld equations require the potential to vanish and the Mij ﬁeld equations require the potential to be stationary, U(M) =

∂U(M) =0. ∂Mij

(8.3.23)

Using an S U(2) × S U(2) rotation to diagonalize M, the conditions (8.3.23) become M11 M22 M33 = M11 M22 = M11 M33 = M22 M33 = 0 .

(8.3.24)

Thus, only one of the diagonal components can be nonvanishing. By an S U(2)×S U(2) rotation we can take this to be M33 . So we do not construct any physically new string backgrounds in this way. A continuous family of static background solutions is known as a ﬂat direction. Here there are nine massless scalars, which by an S U(2) × S U(2) rotation are reduced to the three diagonal ﬁelds, but there is only one ﬂat direction (gauge-equivalent directions not being counted). Of course, we have only computed the cubic term in the potential, and in a general situation this could vanish and higher powers still be nonzero. To establish an exactly ﬂat direction one needs some special argument. Here the special argument is that we can explicitly construct the free CFT for any value of R. Notice that if we treat j 1˜ 1 or j 2˜ 2 as the modulus the world-sheet action looks very complicated, with cosines and sines of X 25 , but the world-sheet theory is solvable because it is equivalent to the free theory obtained by varying R.

8.3 Closed strings and T -duality

247

T -duality From the mass formula n2 w2R2 2 ˜ − 2) , + + (N + N (8.3.25) R2 α2 α we see that as R → ∞ winding states become inﬁnitely massive, while the compact momenta go over to a continuous spectrum. This is as would be expected with a noncompact dimension. Looking at the opposite limit R → 0, however, one gets a surprise. The states with compact momentum become inﬁnitely massive, but the spectrum of winding states now approaches a continuum — it does not cost much energy to wrap a string around a small circle. Thus as the radius goes to zero the spectrum again seems to approach that of a noncompact dimension. This is quite diﬀerent from the behavior in ﬁeld theory, where there is compact momentum n but no winding number w, and no states become light as R → 0. In fact, the R → 0 and R → ∞ limits are physically identical. The spectrum (8.3.25) is invariant under m2 =

α , n↔w . (8.3.26) R This equivalence extends to the interactions as well. Notice that reversing n and w is the same as R → R =

25 p25 L → pL ,

25 p25 R → −pR .

(8.3.27)

Consider the theory at radius R. Recalling the separation X 25 (z, ¯z ) = XL25 (z) + XR25 (¯z ), deﬁne X 25 (z, ¯z ) = XL25 (z) − XR25 (¯z ) .

(8.3.28)

The ﬁeld X 25 has the same OPEs and energy momentum tensor as X 25 , the minus signs always entering in pairs. The only change in the CFT from using X 25 in place of X 25 is that it includes the sign change (8.3.27), which changes the spectrum from that of the theory with radius R to that of the theory with radius R . That is, these are the same theory, one written in terms of X 25 and one in terms of X 25 . It should be noted that the relation between X 25 (z, ¯z ) and X 25 (z, ¯z ) is nonlocal on the world-sheet, because the separate left- and right-moving ﬁelds are nonlocal operators. This equivalence is known as T -duality. That the R → 0 limit is physically the same as the R → ∞ limit is completely unlike the behavior of point particles, and is one more sign that strings see geometry very diﬀerently at short distances. The space of inequivalent theories is the half-line R ≥ α1/2 . We could take instead the range 0 ≤ R ≤ α1/2 but it is more natural to think in terms of the larger of the two equivalent radii:

248

8 Toroidal compactiﬁcation and T -duality

momentum continua are more familiar than winding number continua, and in particular questions of locality are clearer in the larger-R picture. Thus, there is no radius smaller than the self-dual radius R self-dual = RSU(2)×SU(2) = α1/2 .

(8.3.29)

This indication of a minimum distance scale of the order of the string length will recur elsewhere in string perturbation theory, but we will see that nonperturbatively there is structure at shorter distance. It is important in many applications that T -duality acts nontrivially on the string dilaton Φ. Consider the scattering amplitude for gravitons without winding or compact momentum. These states are invariant under T -duality, and so must be the amplitude as well. The latter can be read from the low energy action (8.1.9), and so the 25-dimensional coupling κ25 must be invariant under duality. Since the 26-dimensional coupling is related to this by κ = (2πρ)1/2 κ25 , duality must act as ρ =

α , ρ

κ =

α1/2 κ. ρ

(8.3.30)

Since κ ∝ eΦ , this implies

eΦ =

α1/2 Φ e . ρ

(8.3.31)

T -duality is a symmetry, relating diﬀerent states (backgrounds) in a single theory, and in fact it is a gauge symmetry. We saw that the modulus δR is the 33-component of a (3,3) ﬁeld. A rotation by π around the 1-axis in one of the S U(2)s reverses the sign of this modulus, so decreasing R from the S U(2) × S U(2) radius is gauge-equivalent to increasing it. Thus the Z2 symmetry of duality is a small part of the S U(2) × S U(2) gauge symmetry. This implies further that duality is a symmetry not only of string perturbation theory but of the exact theory. If we have a massless gauge boson to leading approximation, even small explicit violations of the symmetry lead to inconsistency (spontaneous violation is not an issue, as T -duality is already spontaneously broken away from the self-dual radius). This last argument illustrates an important idea, the complementary use of stringy and spacetime arguments. The existence of the enhanced gauge symmetry is a purely stringy phenomenon. We do not have a nonperturbative understanding of string theory, but we do know a great deal about low energy ﬁeld theory, even nonperturbatively, and we can use this.

8.4 Compactiﬁcation of several dimensions 8.4

249

Compactiﬁcation of several dimensions

It is useful to generalize the analysis to k periodic dimensions, Xm ∼ = X m + 2πR , 26 − k ≤ m ≤ 25 .

(8.4.1)

Let d = 26 − k be the number of noncompact dimensions. Spacetime is now M d ×T k . The coordinate periodicity (8.4.1) is being held ﬁxed, but the actual geometry of the k-torus depends on internal metric Gmn . With more than one compact dimension, the antisymmetric tensor also has scalar components Bmn . The total number of scalars from both sources is k 2 . There are also Kaluza–Klein gauge bosons Am µ and antisymmetric tensor gauge bosons Bmµ . The low energy action, obtained as in section 8.1, is

S =

(2πR)k dd x (−Gd )1/2 e−2Φd Rd + 4∂µ Φd ∂µ Φd 2κ20 1 − Gmn Gpq (∂µ Gmp ∂µ Gnq + ∂µ Bmp ∂µ Bnq ) 4 1 1 1 m nµν − Gmn Fµν F − Gmn Hmµν Hnµν − Hµνλ H µνλ , 4 4 12

(8.4.2)

where Φd = Φ − 14 ln det Gmn . The string spectrum The main new issue is the antisymmetric tensor background Bmn . The contribution of this term to the world-sheet Lagrangian density is proportional to Bmn ∂a (g 1/2 ab X m ∂b X n ) ,

(8.4.3)

which is a total derivative for constant Bmn . It thus has no eﬀect locally and the world-sheet theory is still a CFT, a good string background. It does, however, change the spectrum. We will see this in two ways, canonically and via the path integral. In the canonical approach, focus on the zero-mode contribution to the world-sheet action. Inserting X m (σ 1 , σ 2 ) = xm (σ 2 ) + w m Rσ 1

(8.4.4)

into the world-sheet action gives i 1 ˙n + w m w n R 2 ) − Bmn x ˙m w n R . xm x (8.4.5) L = Gmn (˙ 2α α A dot indicates a derivative with respect to world-sheet time σ 2 . The canonical momenta are ∂L 1 pm = − m = (Gmn v n + Bmn w n R) , (8.4.6) ∂v α

250

8 Toroidal compactiﬁcation and T -duality

where v m = i˙ xm . Some unfamiliar signs and is appear because we are using Euclidean world-sheet time. Continuing to Minkowski time, v m becomes the velocity ∂0 xm . The periodicity of the wavefunction implies the quantization of the canonical momenta, pm = nm /R, and so nm vm = α (8.4.7) − Bmn w n R . R The zero-mode contribution to the world-sheet Hamiltonian is 1 Gmn (v m v n + w m w n R 2 ) , (8.4.8) 2α and the closed string mass is 1 2 ˜ − 2) , Gmn (vLm vLn + vRm vRn ) + (N + N 2α2 α = vm ± wmR .

m2 = m vL,R

(8.4.9a) (8.4.9b)

The Bmn background thus shifts the masses of the winding states through ˜ 0 constraint is the dependence of v m on Bmn . The L0 − L ˜ 0 = Gmn (vLm vLn − vRm vRn ) + 4α (N − N) ˜ . = 4α (nm w m + N − N)

(8.4.10)

Alternatively, consider the torus path integral for the partition function. The antisymmetric tensor term is locally a total derivative and so depends only on the topology of the conﬁguration. Consider the conﬁguration X m = (w1m σ 1 + w2m σ 2 )R .

(8.4.11)

The spatial direction on the torus winds w1m times around each periodic spacetime dimension xm , and the timelike dimension w2m times. The Bmn world-sheet action is 2πibmn w1m w2n ,

(8.4.12)

where bmn = Bmn R 2 /α . One can check that the sum over path integral sectors with phase (8.4.12) is related by Poisson resummation to the partition function for the shifted spectrum (8.4.9). In the description above the metric Gmn appears in the action for the world-sheet ﬁelds. For vertex operator calculations it is more convenient to keep the usual action. Write the metric in terms of a tetrad emr , Gmn = emr enr ,

(8.4.13)

with tangent space indices r, s, . . . . The coordinates X r = emr X m then have the standard OPEs. The vertex operator momenta are vmL vmR (8.4.14) krL = erm , krR = erm , α α

8.4 Compactiﬁcation of several dimensions with the inverse tetrad erm . The mass-shell conditions become 1 2 ˜ − 2) , m2 = (krL krL + krR krR ) + (N + N 2 α ˜ . 0 = α (krL krL − krR krR ) + 4(N − N)

251

(8.4.15a) (8.4.15b)

We use the coordinates X r implicitly in any discussion of vertex operators, such as the one that now follows. Narain compactiﬁcation There is an elegant description of the general toroidal compactiﬁcation. Consider the winding state vertex operators eikL ·XL +ikR ·XR . For any given compactiﬁcation, the spectrum of momenta (krL , krR ) forms a lattice in a 2k-dimensional momentum space R2k . That is, the momentum spectrum consists of all integer linear combinations of 2k linearly independent basis vectors. Let us work with the dimensionless momenta lL,R = kL,R (α /2)1/2 and call the corresponding lattice Γ. The OPE of two vertex operators is

: eikL ·XL (z)+ikR ·XR (¯z ) : : eikL ·XL (0)+ikR ·XR (0) : ∼ z lL ·lL ¯z lR ·lR : ei(kL +kL )·XL (0)+i(kR +kR )·XR (0) : .

(8.4.16)

As one vertex operator circles the other, the product picks up the phase exp[2πi(lL · lL − lR · lR )]. Single-valuedness of the operator product thus requires lL · lL − lR · lR ≡ l ◦ l ∈ Z

(8.4.17)

l

for all l and in Γ. We have deﬁned the product ◦, which has signature (k, k) in R2k . The dual lattice Γ∗ is deﬁned to be the set of points in R2k that have integer ◦ product with all points in Γ. The single-valuedness condition (8.4.17) is then the statement that Γ ⊂ Γ∗ .

(8.4.18)

Modular invariance also constrains Γ. In section 7.2 it was shown that ˜ 0 be an integer for all invariance under τ → τ + 1 requires that L0 − L states in the string CFT. Then l ◦ l ∈ 2Z for all l ∈ Γ .

(8.4.19)

The condition (8.4.19) actually implies condition (8.4.17), given closure of the OPE 2l ◦ l = (l + l ) ◦ (l + l ) − l ◦ l − l ◦ l ∈ 2Z .

(8.4.20)

Modular invariance under τ → −1/τ requires a more detailed calculation. The partition function for the compact dimensions is ZΓ (τ) = |η(τ)|−2k

l∈Γ

exp(πiτlL2 − πi¯τlR2 ) .

(8.4.21)

252

8 Toroidal compactiﬁcation and T -duality

To determine the transformation property of ZΓ use again Poisson resummation. Write

δ(l − l ) = VΓ−1

l ∈Γ

exp(2πil ◦ l) .

(8.4.22)

l ∈Γ∗

Here, VΓ is the volume of a unit cell of the lattice Γ. Eq. (8.4.22) is familiar from Fourier series: the phase averages to zero except when l ∈ Γ, and the normalization is ﬁxed by integrating over a unit cell. Using the representation (8.4.22), the sum (8.4.21) can be written ZΓ (τ) = VΓ−1 |η(τ)|−2k =

l ∈Γ∗ −1 −k/2 VΓ (τ¯τ) |η(τ)|−2k

d2k l exp(2πil ◦ l + πiτlL2 − πi¯τlR2 )

exp(−πilL2 /τ + πilR2 /¯τ)

l ∈Γ∗

= VΓ−1 ZΓ∗ (−1/τ) ,

(8.4.23)

where in the last line we have used the modular transformation (7.2.44) of the eta function. Then the condition Γ = Γ∗

(8.4.24)

is suﬃcient for modular invariance, because VΓ = VΓ−1 ∗ and so is equal to unity if (8.4.24) holds. A little thought shows that this is also necessary if modular invariance is to hold for all τ. The consistency conditions can be summarized as the requirement that Γ be an even self-dual lattice of signature (k, k), even referring to the property (8.4.19) and self-dual to the property (8.4.24). All such lattices have been classiﬁed. Note that the consistency conditions (8.4.19) and (8.4.24) depend on the momenta l only through the ◦ product, which is invariant under Lorentz boosts of the 2k-dimensional space, O(k, k, R) transformations. If Γ is an even self-dual lattice then so is Γ = ΛΓ ,

(8.4.25)

which is obtained by rotating all the points of the lattice by a common O(k, k, R) transformation Λ. Now, it is important that O(k, k, R) is not a symmetry of the theory. The mass-shell condition and operator products involve the separate dot products lL · lL and lR · lR and so are invariant only under the O(k, R) × O(k, R) that acts on lL and lR separately. Thus, most O(k, k, R) transformations produce inequivalent theories. Consider for example k = 1, where n mr (8.4.26) lL,R = ± r 2

8.4 Compactiﬁcation of several dimensions

253

in terms of the dimensionless radius r = R(2/α )1/2 . This is indeed an even self-dual lattice. The boost lL = lL cosh λ + lR sinh λ ,

lR = lL sinh λ + lR cosh λ

(8.4.27)

changes this to the lattice for r = re−λ . All even self-dual lattices of given Lorentzian signature can be obtained from any single lattice by O(k, k, R) transformations. Start with a given solution, for example all compact dimensions orthogonal and at the S U(2) radius with Bmn = 0. The corresponding momentum lattice Γ0 is known as kP2 and gives the gauge group S U(2)2k . As discussed above, two O(k, k, R) transformations Λ and Λ Λ give equivalent string theories if Λ ∈ O(k, R) × O(k, R), so the space of inequivalent theories obtained in this way is O(k, k, R) , O(k, R) × O(k, R)

(8.4.28)

up to some discrete identiﬁcations to be discussed shortly. This is equivalent to the earlier description in terms of the backgrounds of Gmn and Bmn . In particular, the number of parameters in the coset (8.4.28) is 2k(2k − 1) (8.4.29) − k(k − 1) = k 2 , 2 the same as the number of components in Gmn and Bmn . The T -duality symmetry is much enlarged from the case of one compactiﬁed dimension. In the abstract description, there is some discrete subgroup of O(k, k, R) that takes the initial lattice Γ0 into itself (though it permutes the individual points of the lattice). This subgroup is conventionally but somewhat inaccurately denoted O(k, k, Z). If Λ is in this group, then clearly ΛΓ0 and ΛΛ Γ0 are the same lattice. In all we have the equivalence ΛΓ0 ∼ = Λ ΛΛ Γ0 , Λ ∈ O(k, R) × O(k, R) ,

Λ ∈ O(k, k, Z) .

(8.4.30a) (8.4.30b)

The space of inequivalent lattices, and inequivalent backgrounds, is then O(k, k, R) . O(k, R) × O(k, R) × O(k, k, Z)

(8.4.31)

One should remember that the continuous denominator group acts by left-multiplication and the discrete group by right-multiplication, as in eq. (8.4.30a). In terms of the backgrounds, the T -duality group O(k, k, Z) contains transformations of several types. One is R → α /R dualities on the individual axes. Another is large spacetime coordinate transformations respecting

254

8 Toroidal compactiﬁcation and T -duality

the periodicity, xm = Lmn xn

(8.4.32)

with Lmn integers and det L = 1 for invertibility. This is the group S L(k, Z). Finally there are shifts of the antisymmetric tensor background, bmn → bmn + Nmn

(8.4.33)

with Nmn integers. From either the canonical result (8.4.7) or the path integral phase (8.4.12) it follows that this leaves the spectrum invariant. Together these generate the whole T -duality group. Identifying ﬁrst by the left-multiplication produces the space (8.4.28), and one can think of the discrete right-multiplication as acting on this covering space. For generic Λ, there are no Λ1,2 in the respective denominator groups such that Λ1 ΛΛ2 = Λ, so that there are no T -dualities that leave this point on the covering space ﬁxed. At special Λ there are solutions to Λ1 ΛΛ2 = Λ, and these will be ﬁxed points of the T -duality element Λ2 . In the one-dimensional case we had such a ﬁxed point, which was also a point of enhanced gauge symmetry. In the higher-dimensional case, besides products of S U(2)s, larger gauge groups arise at various points. We will defer further discussion of this to chapter 11. When there is a moduli space parameterized by scalar ﬁelds φi , the kinetic term 1 − gij (φ)∂µ φi ∂µ φj 2

(8.4.34)

deﬁnes a natural metric on moduli space. To be precise, an unambiguous deﬁnition requires that one ﬁrst perform a spacetime Weyl transformation as in eq. (3.7.25) so as to make the coeﬃcient of the gravitational action moduli-independent and eliminate mixing between the moduli and the spacetime metric. In the low energy action (8.4.2), the resulting scalar kinetic terms are proportional to 16 ∂µ Φ∂µ Φ + Gmn Gpq (∂µ Gmp ∂µ Gnq + ∂µ Bmp ∂µ Bnq ) . d−2

(8.4.35)

The coset space (8.4.31) has a unique O(k, k, R)-invariant metric, which is just the second term in eq. (8.4.35). This O(k, k, R) is not a symmetry of the full theory; only its discrete T -duality subgroup O(k, k, Z) is. The diﬀerence between the two is not visible at low energy — it comes from the quantization of the string zero modes and so aﬀects the massive spectrum. Thus O(k, k, R) is an accidental symmetry of the low energy theory. In this example the moduli space is a product of the dilaton moduli space and the moduli space of compactiﬁcations.

8.4 Compactiﬁcation of several dimensions

255

An example The case of two compact dimensions is a good example. The four moduli G24,24 , G24,25 , G25,25 , and B24,25 are usefully joined into two complex ﬁelds τ = τ1 + iτ2 and ρ = ρ1 + iρ2 . Deﬁne R2 B24,25 + i det 1/2 Gmn α V = b24,25 + i 2 , (8.4.36) 4π α where V is the volume of the compact two-torus. Parameterize the compact metric as 2 α ρ2 ds2 = 2 dX 24 + τdX 25 . (8.4.37) R τ2

ρ=

The coordinate transformations (8.4.32) act on the spacetime two-torus just as the modular group acts on the world-sheet two-torus, generating a P S L(2, Z) on τ and leaving ρ invariant. The antisymmetric tensor shift (8.4.33) is ρ → ρ + N24,25 with τ invariant. A simultaneous T -duality on X 24,25 acts as ρ → −1/ρ with τ invariant. Together the last two transformations generate a second P S L(2, Z), acting on ρ. In addition, duality on X 24 alone takes (τ, ρ) → (ρ, τ) and spacetime parity X 24 → −X 24 takes (τ, ρ) → (−¯τ, −¯ ρ). The full T -duality group is then P S L(2, Z) × P S L(2, Z)× Z22 .

(8.4.38)

A further equivalence comes from world-sheet parity, which takes (τ, ρ) → (τ, −¯ ρ). It reverses the sign of the ◦ product. The moduli space is therefore two copies of the moduli space of the torus, together with some discrete identiﬁcations. The kinetic terms are proportional to ¯ ∂µ τ∂µ¯τ ∂µ ρ∂µ ρ + . 2 2 τ2 ρ2

(8.4.39)

The modular-invariant combination d2 τ/τ22 was encountered in the discussion of the genus-zero amplitude. The continuous group P S L(2, R) carries any τ in the upper half-plane into any other, and a U(1) subgroup leaves each τ invariant. Thus the upper half-plane can be regarded as P S L(2, R)/U(1), and the full moduli space is P S L(2, R) × P S L(2, R) . U(1) × U(1) × P S L(2, Z) × P S L(2, Z) × Z22

(8.4.40)

This is the same as the earlier (8.4.31). In particular, P S L(2, R)×P S L(2, R) is locally the same as O(2, 2, R), in parallel to the fact that S U(2) × S U(2) is locally the same as O(4, R).

256

8 Toroidal compactiﬁcation and T -duality 8.5

Orbifolds

Rather than the periodic identiﬁcation of X 25 , consider instead identifying points under the reﬂection X 25 ∼ = −X 25 .

(8.5.1)

Now a fundamental region is the half-line X 25 ≥ 0, and the hyperplane X 25 = 0 becomes a boundary. This hyperplane is distinguished by the fact that its points are ﬁxed under the reﬂection (8.5.1). More generally we could identify spacetime under the simultaneous reﬂection of k coordinates X m → −X m ,

26 − k ≤ m ≤ 25 .

(8.5.2)

Again there is a space of ﬁxed points, at X 26−k = . . . = X 25 = 0. For k ≥ 2 this space is a singularity. In particular, for k = 2 it is a conic singularity with deﬁcit angle π. One can form a compact space by combining the reﬂection (8.5.1) with the periodic identiﬁcation X 25 ∼ = X 25 + 2πR; we will call these r and t respectively. One can think of ﬁrst identifying by t to form the circle S 1 , and then identifying points on the circle under r. The resulting compact space is a line segment, with 0 ≤ X 25 ≤ πR as a fundamental region. There are ﬁxed points at both ends of the segment, the point πR being ﬁxed by tr. In all, one is identifying the real line under the group tm : tm r :

X 25 X 25

∼ = X 25 + 2πRm , ∼ = 2πRm − X 25 ,

(8.5.3a) (8.5.3b)

for any integer m. Similarly one could identify the k-torus under the reﬂection (8.5.2). In that case there are 2k distinct ﬁxed points, with each of the X m being either 0 or πR. The singular spaces obtained in this way are known as orbifolds. The space obtained by the identiﬁcation (8.5.2) in noncompact space is Rk /Z2 , while the compact space obtained by the same identiﬁcation on the k-torus is T k /Z2 . It is not obvious that string theory on such singular spaces should make sense, but we will see that it does. It is not so diﬀerent from toroidal compactiﬁcation, but it breaks more symmetry (and, later on, supersymmetry) and so will be of interest in constructing string models. The identiﬁcation (8.5.1) or (8.5.2) has two eﬀects. The ﬁrst is that the string wavefunction must be invariant under the reﬂection, equal at identiﬁed points. Second, there is a new sector in the closed string spectrum, in which X 25 (σ 1 + 2π) = −X 25 (σ 1 ) ,

(8.5.4)

257

8.5 Orbifolds

since these are the same point in the identiﬁed spacetime. Strings in this sector are known as twisted states. This is directly parallel to periodic identiﬁcation. There, invariance of the wavefunction made the momentum discrete, while strings that closed only up to the periodic identiﬁcation (winding strings) appeared. One can again make a pictorial argument, analogous to ﬁgure 8.1, to show that twisted strings can be produced from untwisted ones. We will also see below that modular invariance requires them in the spectrum. We will focus on the compact one-dimensional orbifold S 1 /Z2 . Examining ﬁrst the untwisted sector, the spectrum of the periodic theory is reduced by the projection onto invariant states. The eﬀect of r on a general state is ˜ k µ , n, w → (−1) |N, N;

∞ m=1

(Nm25 +N˜ m25 ) |N, N; ˜ k µ , −n, −w ,

(8.5.5)

in particular reversing the compact winding and momentum. We must form linear combinations that are invariant under this. The states that are massless at generic R have n = w = 0, so the projection simply requires that the number of 25-excitations be even. The spacetime graviton, antisymmetric tensor, dilaton, and tachyon survive this projection. The µ α25 modulus α25 −1 ˜ −1 |0; k , 0, 0 also survives — this could have been anticipated, since the radius R can take any value. However, the Kaluza–Klein gauge bosons are no longer in the spectrum. The fate of the extra massless states at the self-dual point is quite interesting, and we will discuss it in detail later. In the sector twisted by r, X 25 is antiperiodic and so has a half-integral mode expansion 25

X (z, ¯z ) = i

1/2 ∞ α

2

α25 α˜25 1 m+1/2 m+1/2 + m+1/2 m+1/2 m + 1/2 ¯ z z m=−∞

.

(8.5.6)

The antiperiodicity forbids any center-of-mass coordinate or momentum, so the string cannot move away from the X 25 = 0 ﬁxed point: to the extent that the oscillations are small, one has X 25 (z, ¯z ) ≈ 0. There are also twisted strings localized at the other ﬁxed point, X 25 (σ 1 + 2π) = 2πR − X 25 (σ 1 ) .

(8.5.7)

The ﬁelds change by tr as one circles the loop. The mode expansion is as above with an additional constant term πR. All other ﬁxed points nπR are images of one of these under t. According to the zero-point mnemonic from section 2.9, the zero-point 1 1 for a periodic boson to 48 for an energy of X 25 is shifted from − 24 1 antiperiodic boson, for a net + 16 . The mass-shell conditions in the twisted

258

8 Toroidal compactiﬁcation and T -duality

sectors are then

4 15 ˜ . m = N− , N=N (8.5.8) α 16 Also, the 25-oscillators make half-integral contributions to the level N. The r-projection again requires the total number of 25-excitations to be even, ˜ In which can also be deduced from the level-matching condition N = N. the twisted sectors the ground states are |T1,2 , the subscript referring to the two ﬁxed points. They are tachyonic, as are the ﬁrst excited states α25 α25 −1/2 ˜ −1/2 |T1,2 . There are no massless twisted states. The extension to the more general case (8.5.2) is straightforward. The quantization is perfectly consistent in spite of the singularities in spacetime. The one complication with orbifolds is that the twisted state vertex operators are not as simple as those for winding states; there is no such explicit formula as the exponential of a free ﬁeld. For two twisted strings and any number of untwisted ones, the tree-level amplitude can be written in operator form T |VU VU . . . |T , which uses only untwisted vertex operators. For four or more twisted vertex operators (the number must be even for the path integral to make sense) two methods have been used. One is the stress tensor method, which we used in section 7.2 to get the partition function on the torus. The other is to go to the covering space on which X 25 is single-valued (this is a genus g surface for 2g + 2 twisted vertex operators on the sphere) and evaluate the path integral. There is a simple result for the tree-level couplings of untwisted states. On the sphere, with all external states untwisted, the twists and projections do not enter into the calculation at all and the amplitudes are the same as in the untwisted theory. For example, the low energy eﬀective action for the massless untwisted ﬁelds is the same as we found for toroidal compactiﬁcation, with the omission of the vector ﬁelds. This property is often useful in analyzing the physics of orbifold theories; it is called the inheritance principle. Consider now the partition function of the X 25 CFT. In the untwisted sector, one is to project the spectrum onto r = +1 states, 1 + r L0 L˜ 0 . (8.5.9) (q¯ q )−1/24 TrU q q¯ 2 The ﬁrst term in the projection operator just gives half the partition function (8.2.9) for toroidal compactiﬁcation. In the term with r in the trace, the diagonal elements of r must have n = w = 0, as one sees from 25 ˜ 25 eq. (8.5.5). The (−1)N +N changes the oscillator sum to 1 − q + q 2 − q 3 + . . . = (1 + q)−1 . The partition function for the untwisted states is then 2

∞ 1 1 |1 + q m |−2 . Ztor (R, τ) + (q¯ q )−1/24 2 2 m=1

(8.5.10)

259

8.5 Orbifolds

The contribution of the twisted sectors is from the sum over half-integer moded oscillations, 1 + r L0 L˜ 0 (q¯ q )1/48 TrT q q¯ 2 = (q¯ q)

1/48

∞ ∞ −2 −2 m−1/2 m−1/2 , 1 − q + 1 + q m=1

(8.5.11)

m=1

the factor of 2 from the number of twisted sectors oﬀsetting the 12 in the projection operator. In all, the partition function can be written as η(τ) η(τ) η(τ) 1 . (8.5.12) + + Zorb (R, τ) = Ztor (R, τ) + 2 ϑ10 (0, τ) ϑ01 (0, τ) ϑ00 (0, τ) The ﬁrst term is already known to be modular-invariant, and it follows from the modular transformation properties given in section 7.2 that the sum of the ﬁnal three is invariant as well. In terms of the path integral on the torus, Ztor (R, τ) is from ﬁelds that are periodic up to translations. The term with ϑab is from the path integral with X 25 (σ 1 + 2π, σ 2 ) = (−1)a+1 X 25 (σ 1 , σ 2 ) , X 25 (σ 1 + 2πτ1 , σ 2 + 2πτ2 ) = (−1)b+1 X 25 (σ 1 , σ 2 ) .

(8.5.13a) (8.5.13b)

For example, the ϑ10 term is from the untwisted sector with r in the trace, that is, from ﬁelds that are periodic in the spatial direction and antiperiodic in the time direction. The ϑ01 term, from the twisted sector with 1 in the trace, has the opposite boundary conditions. These are interchanged by τ → −1/τ. Thus, the twisted sectors are necessary for modular invariance. Twisting Toroidal and orbifold compactiﬁcations are two examples of a useful construction, known as twisting, for constructing new string theories from old. In a given CFT whose symmetries include some discrete group H, we can form a new CFT in two steps. First add twisted sectors, in which the closed string world-sheet ﬁelds are periodic only up to some transformation h ∈ H, φ(σ 1 + 2π) = h · φ(σ 1 ) ,

(8.5.14)

where φ denotes a generic world-sheet ﬁeld. Then, restrict the spectrum to H-invariant states. Although the ﬁelds (8.5.14) are no longer periodic, the vertex operators are periodic because of the projection. So the OPE is still well deﬁned, and it closes because the product of H-invariant operators is H-invariant.

260

8 Toroidal compactiﬁcation and T -duality

In the partition function, we sum over twists h1 in the spatial direction and include a projection operator onto invariant states, 1 hˆ 2 . PH = (8.5.15) order(H) h ∈H 2

The operator hˆ 2 in the trace makes the ﬁelds in the time direction aperiodic by h2 , so in the end we are summing over twists in both the spatial and time directions on the torus, 1 Z= Zh ,h . (8.5.16) order(H) h ,h ∈H 1 2 1

2

The sum over boundary conditions is modular-invariant after a shift of summation variables. The modular transformation τ → −1/τ takes (h1 , h2 ) to (h2 , h−1 1 ), and τ → τ + 1 takes (h1 , h2 ) to (h1 , h1 h2 ). Thus, if we project onto H-invariant states by summing over h2 , we must also sum over the twisted sectors. In more general situations, in particular right–left asymmetric theories, there will be an issue of phases possibly spoiling the naive modular invariance of the sum over boundary conditions, but we will deal with that in chapters 10, 11, and 16. In the case of toroidal compactiﬁcation, the original CFT is the noncompact theory and H consists of tm = exp(2πiRmp) for all integer m. This happens to have inﬁnite order, but we can regulate the sum by putting the system in a large box. For the orbifold, we can think of starting from the toroidal theory and twisting by r, or from the noncompact theory and twisting by r and t. When H is non-Abelian, the path integral boundary conditions are inconsistent for h1 and h2 noncommuting and the path integral vanishes. This also follows from the canonical expression (8.5.15). If φ has spatial twist h1 , then φ (σ 1 ) = h2 · φ(σ 1 )

(8.5.17)

φ (σ 1 + 2π) = h1 · φ (σ 1 ) ,

(8.5.18)

h1 = h2 h1 h−1 2

(8.5.19)

has a diﬀerent spatial twist where and so the diagonal matrix elements of hˆ 2 are zero. We also see that the sectors twisted by h1 and h1 are not independent — they are related by the projection onto h2 -invariant states. Thus the independent twisted sectors are in one-to-one correspondence with the conjugacy classes of H. Twisting can also be thought of as gauging the discrete group H. It may not be obvious that this makes sense, as the discreteness forces the gauge

8.5 Orbifolds

261

parameter to be constant. In particular, there can be no associated gauge ﬁeld. The point is that on a world-sheet with nontrivial closed paths, only gauge-invariant quantities need to be periodic, so the ﬁelds can be aperiodic by an H transformation. Gauging a discrete symmetry adds in these path integral sectors and correspondingly projects onto invariant states. This is just the twisting construction. It is also possible to gauge a continuous world-sheet symmetry to produce a new CFT. This is more involved and will be discussed in chapter 15. The inheritance principle holds for general twisted theories: the path integral with untwisted vertex operators on the sphere is unaﬀected by the twist. c = 1 CFTs We have found two families of compact c = 1 CFTs. These are toroidal compactiﬁcation, restricting the radius to R ≥ α1/2 so as to count only distinct theories, and orbifold compactiﬁcation with the same range of R. These two families are connected. Starting from the S U(2) × S U(2) radius R = α1/2 , twist by r :

X 25 → X 25 + πα1/2 .

(8.5.20)

This produces the toroidal theory at R = α1/2 /2, which by T -duality is the same as at R = 2α1/2 . In terms of the S U(2) currents (8.3.12), r ﬂips the signs of j 1,2 and ˜ 1,2 and so is a rotation by π around the 3-axis in each S U(2). The orbifold r ﬂips the signs of j 2,3 and ˜ 2,3 and is a rotation by π around the 1-axis in each S U(2). But of course these rotations are equivalent under the S U(2) × S U(2), so the resulting theories are the same, the R = 2α1/2 torus and the R = α1/2 orbifold. The corresponding partition functions are equal, Zorb (α1/2 , τ) = Ztor (2α1/2 , τ) ,

(8.5.21)

as can be veriﬁed by theta function identities. This equivalence holds only at these radii. For example, at generic R the toroidal theory has a U(1) × U(1) gauge symmetry while the orbifold has no gauge symmetry. Thus, the two moduli spaces are connected as shown in ﬁgure 8.2. This structure, the moduli space of vacua having distinct branches meeting at special points, will be a very common and important feature in supersymmetric theories. On each branch there is in general a distinct low energy eﬀective ﬁeld theory, here diﬀering in the low energy gauge symmetry. In the neighborhood of the special point there are more light ﬁelds, diﬀerent subsets of which become massive as we move away along the various branches.

262

8 Toroidal compactiﬁcation and T -duality

Rorb α 1/2 D4 4

T

O

I

D3 3

D2 2

Rtor

SU(2) 2 α 1 2 1/2 1/2

Z2 2

Z3 3

Z4 4

Z5 5

Fig. 8.2. Moduli space of c = 1 CFTs. The toroidal branch is horizontal and the orbifold branch is vertical. The special points obtained by twisting by a discrete subgroup of SU(2) are indicated, including the three isolated points obtained from the tetrahedral, octahedral, and icosahedral groups. Other special radii, such as the torus at R = (2α )1/2 , will arise later.

It is instructive to analyze the low energy physics near the meeting of the toroidal and orbifold branches. Describe this point in terms of twisting the S U(2) theory by r , eq. (8.5.20). The massless scalars surviving the twist are j 3˜ 3 and j i˜ j for i, j ∈ {1, 2}. Since these are all untwisted, the potential is the same as before twisting, namely det M. Again the scalar ﬁeld Mij can be diagonalized, by a U(1) × U(1) rotation, and the potential is ﬂat when exactly one diagonal element is nonzero. Now, however, the modulus M33 cannot be rotated into M11 or M22 so these directions are physically distinct. The modulus M33 leaves the U(1) × U(1) unbroken and corresponds to moving along the toroidal branch. The modulus M11 or M22 breaks the gauge symmetry and corresponds to moving along the orbifold branch. Since this modulus is charged, a reversal of its sign is a gauge transformation and so the line of physically distinct orbifolds ends at the point where the branches meet. We cannot simultaneously vary both moduli, because the linear combination is not a ﬂat direction of the potential. That string theory on an orbifold can be equivalent to string theory on a circle is quite remarkable, one more indication that the string does not see spacetime geometry in the same way that we do. Moving along the toroidal branch, there are extra massless states at all multiples of the S U(2) radius, R = kα1/2 . These are the states with

263

8.6 Open strings

(n, w) = (±2k, 0). We can think of these points as obtained, in the T -dual description R = α1/2 /k, by twisting the S U(2) × S U(2) theory by the Zk symmetry 2πα1/2 . (8.5.22) k This multiplies j 1 + ij 2 and ˜ 1 + i˜ 2 by exp(2πi/k) and so the surviving massless scalars are X 25 → X 25 +

j 3˜ 3 ,

j 1˜ 1 + j 2˜ 2 ,

j 1˜ 2 − j 2˜ 1 .

(8.5.23)

The ﬁrst of these changes the radius of the torus and so is a ﬂat direction as always. However, there are no other branches emerging from these points, because the conditions (8.3.23) on the potential for a ﬂat direction are not met in the directions M11 = M22 or M12 = −M21 . The shift (8.5.22) generates a Zk subgroup of S U(2) × S U(2). Let us consider twists by other subgroups. The dihedral group Dk generated by the shift (8.5.22) and the reﬂection X 25 → −X 25 produces the orbifold at radius kα1/2 . The other three discrete subgroups are the tetrahedral, octahedral, and icosahedral groups. These remove all moduli from the CFT, so the twisted CFTs are isolated points, not members of continuous families. Unlike the Zk and Dk twists, which can be deﬁned at all radii, these discrete groups involve elements of S U(2) that exist only at the critical radius and so the radius cannot be varied after twisting. String theory with such a compact space would have only one scalar, the dilaton. These are all of the known c = 1 CFTs, and so give all known 25dimensional bosonic string backgrounds. 8.6

Open strings

The new feature in toroidal compactiﬁcation of the open string is the possibility of a nontrivial Wilson line, a ﬂat background for the gauge ﬁeld. Studying ﬁrst a U(1) gauge theory with some charged ﬁelds, consider the constant background

∂Λ θ iθx25 , (8.6.1) = −iΛ−1 25 , Λ(x25 ) = exp − A25 (x ) = − 2πR ∂x 2πR where θ is a constant. Locally this is pure gauge. The ﬁeld strength vanishes and the ﬁeld equations are trivially satisﬁed. However, the gauge parameter Λ does not satisfy the spacetime periodicity, and so the background has a physical eﬀect. The gauge-invariant quantity that measures this is the Wilson line M

Wq = exp iq

25

dx A25 = exp(−iqθ) .

(8.6.2)

264

8 Toroidal compactiﬁcation and T -duality

Consider ﬁrst a point particle of charge q, with gauge-ﬁxed world-line action 1 ˙M ˙ m2 ˙M . S = dτ (8.6.3) X XM + − iqAM X 2 2

The gauge action is simply −iq dxM AM , so paths that wind around the compact dimension pick up a phase just equal to the Wilson line Wq . The canonical momentum is ∂L qθ p25 = − 25 = v 25 − , (8.6.4) ∂v 2πR ˙ 25 as in eq. (8.4.6). The wavefunctions must be periodic in where v 25 = iX the compact space, so p25 = l/R for integer l and 2πl + qθ . (8.6.5) 2πR The Hamiltonian, which annihilates physical states, is 1 2 H = (pµ pµ + v25 + m2 ) , (8.6.6) 2 so the masses −pµ pµ are shifted due to the dependence of v25 on θ. The same spectrum is readily obtained in a ﬁeld rather than particle description. Note that v25 is just the gauge-invariant momentum −i∂25 − qA25 . One can also make the gauge transformation Λ−1 to set A25 to zero. Charged ﬁelds in this gauge are no longer periodic, picking up a phase exp(iqθ) under x25 → x25 + 2πR. Physical (gauge-invariant) objects are still periodic. The canonical momentum is now shifted, and we have 2πl + qθ v25 = p25 = (8.6.7) 2πR with the same result as before for the gauge-invariant momentum v25 . Let us turn now to the string, and include U(n) Chan–Paton factors. A general constant A25 can be diagonalized by a gauge transformation, 1 diag (θ1 , θ2 , . . . , θn ) . A25 = − (8.6.8) 2πR The gauge ﬁeld is in the diagonal subgroup of U(n), namely U(1)n . The gauge ﬁeld couples to the Chan–Paton factor of a general state as [ AM , λ ], so a string in Chan–Paton state |ij has charge +1 under U(1)i , −1 under U(1)j , and is neutral under the others. Thus it has v25 =

2πl − θj + θi . 2πR The open string spectrum is then v25 =

m2 =

(2πl − θj + θi )2 1 + (N − 1) . 2 2 4π R α

(8.6.9)

(8.6.10)

265

8.6 Open strings

Consider in particular the gauge bosons, where l = 0 and N = 1 and so (θj − θi )2 . (8.6.11) 4π 2 R 2 In generic backgrounds, all the θs are distinct and the only massless vectors are the diagonal ones, i = j. The unbroken gauge group in this case is U(1)n . If r of the θs are equal, the corresponding r × r matrix of vectors is massless, carrying the gauge symmetry U(r). With the n θs equal in sets of ri , the gauge symmetry is m2 =

U(r1 ) × . . . × U(rs ) ,

s

ri = n .

(8.6.12)

i=1

As always, there is a low-energy interpretation. The gauge ﬁeld A25 is a 25dimensional scalar in the adjoint representation of U(n), and its vacuum expectation value breaks the symmetry spontaneously to U(r1 )×. . .×U(rs ). The reader can ﬁll in the details of the eﬀective low-energy action. It is important that with k compact dimensions, the potential contains a term

Tr [Am , An ]2

(8.6.13)

from the ﬁeld strength in the 26-dimensional Yang–Mills action. This forces the gauge ﬁelds in diﬀerent directions to commute in a static solution, so that they are simultaneously diagonalizable. There are then kn moduli from the gauge ﬁeld and k 2 from the metric and antisymmetric tensor. T -duality Now consider the R → 0 limit of the open string spectrum. Open strings with Neumann boundary conditions have no quantum number comparable to w; they can always be unwound from the periodic dimension. So when R → 0 the states with nonzero momentum go to inﬁnite mass, but there is no new continuum of states. The behavior is as in ﬁeld theory: the resulting states move in 25 spacetime dimensions. A seeming paradox arises when one remembers that theories with open strings always have closed strings as well, so that in the R → 0 limit the closed strings move in 26 spacetime dimensions but the open strings only in 25. One can reason out what is happening as follows. The interior of the open string is made of the same ‘stuﬀ’ as the closed string, and so should still be vibrating in 26 dimensions. What distinguishes the open string is its endpoints, so it is these that must be restricted to a 25-dimensional hyperplane. Indeed, recall that one describes the T -dual theory by using the new

266

8 Toroidal compactiﬁcation and T -duality

Fig. 8.3. Open strings with endpoints attached to a hyperplane. The dashed planes are periodically identiﬁed. Two strings are shown, with winding numbers zero and one.

embedding coordinate X 25 (z, ¯z ) = XL25 (z) − XR25 (¯z ) .

(8.6.14)

∂n X 25 = −i∂t X 25 ,

(8.6.15)

Then

with n the normal and t the tangent at the boundary. The Neumann condition on the original coordinate is a Dirichlet condition on the dual coordinate: the X 25 coordinate of each string endpoint is ﬁxed. Let us ﬁrst consider compactiﬁcation without Wilson lines. Then all endpoints are actually constrained to lie on the same hyperplane. To see this, integrate X

25

(π) − X

25

π

(0) = 0

1

dσ ∂1 X

= −2πα v 25

25

= −i

π 0

dσ 1 ∂2 X 25

2πα l =− = −2πlR . R

(8.6.16)

The total change in X 25 between the two ends is an integral multiple of the periodicity 2πR of the dual dimension, so the ends lie on the same hyperplane in the periodic T -dual space. For two diﬀerent open strings, consider the connected world-sheet that results from graviton exchange between them. One can carry out the same argument (8.6.16) on a path connecting any two endpoints, so all endpoints of all strings lie on the

267

8.6 Open strings

0

θ 1R'

θ3R' 2 π R'

θ 2 R'

Fig. 8.4. n = 3 hyperplanes at diﬀerent positions, with various strings attached. Ignoring the tachyon, the lightest strings with both ends on the same hyperplane are massless, while strings stretched between noncoincident hyperplanes are massive due to their tension. When two hyperplanes become coincident, the lightest strings stretched between them become massless.

same hyperplane, as in ﬁgure 8.3. The ends are still free to move in the other 24 spatial dimensions. Now we consider the eﬀect of Wilson lines. Due to the shift (8.6.9) in v25 , the diﬀerence (8.6.16) is replaced by ∆X 25 = X 25 (π) − X 25 (0) = −(2πl − θj + θi )R .

(8.6.17)

In other words, up to an arbitrary additive normalization, the endpoint in state i is at X 25 = θi R = −2πα A25,ii . (8.6.18) Thus there are in general n hyperplanes at diﬀerent positions as depicted in ﬁgure 8.4. The mode expansion in the sector with Chan–Paton state |ij , for an open string winding l times around the compact dimension, is X 25 (z, ¯z ) = θi R − = θi R +

α iR (2πl − θj + θi ) ln(z/¯z ) + i 2π 2

1/2 ∞ m=−∞ m=0

α25 m (z −m − ¯z −m ) m

∞ α25 σ1 m ∆X 25 − (2α )1/2 exp(−mσ 2 ) sin mσ 1 . π m m=−∞ m=0

(8.6.19)

268

8 Toroidal compactiﬁcation and T -duality

The spectrum (8.6.10) becomes 2

m =

∆X 25 2πα

2

+

1 (N − 1) , α

(8.6.20)

where ∆X 25 , given by eq. (8.6.17), is the minimum length for a string in the given sector. Now consider how this picture generalizes if several coordinates are periodic, say X m for p + 1 ≤ m ≤ 25, and we rewrite the periodic dimensions in terms of the dual coordinate. The Neumann conditions on X m again become Dirichlet conditions for the dual coordinates X m , so the open string endpoints are conﬁned to n parallel (p + 1)-dimensional subspaces. The Wilson lines in the various directions become independent coordinates for each subspace. The dual picture is rather odd in that translation invariance is broken by the open string boundary conditions. This simply reﬂects the fact that in the original open string theory winding number is not conserved (a closed winding string can break open and unwind) and winding number is T -dual to momentum. It is important to remember that T -duality is just a diﬀerent description of the same theory, and further that when the compactiﬁcation radii in the original theory are small the T -dual picture is the natural one to use. Figure 8.4 then shows how to interpret the resulting physics. 8.7

D-branes

We have just learned that when an open string theory is compactiﬁed on a small torus, the physics is described by compactiﬁcation on a large torus but with the open string endpoints restricted to lie on subspaces. In fact, these subspaces are new dynamical objects in their own right. Consider the spectrum of massless open strings for the generic conﬁguration in which all θi are distinct, taking for simplicity the case in which only one coordinate is dualized. Ignoring the tachyon, the mass (8.6.20) vanishes only for N = 1 and ∆X 25 = 0. That is, both endpoints are on the same hyperplane, with zero winding. We have therefore the massless states αµ−1 |k; ii ,

V = i∂t X µ ,

α25 −1 |k; ii

V = i∂t X

,

25

(8.7.1a) = ∂n X

25

.

(8.7.1b)

These are of course the same massless states as in the original theory, T -duality simply giving a diﬀerent picture of the same spectrum. For the generic Wilson line, the only massless open string states in the original theory would have been the n massless U(1) vectors. In eq. (8.7.1) we

8.7 D-branes

269

have separated these according to whether the polarization is tangent or perpendicular to the hyperplane. The 25 states with tangent polarization comprise a gauge ﬁeld living on the hyperplane. The state with perpendicular polarization was the gauge ﬁeld in the compact direction of the original theory, and it has a simple and important interpretation in the T -dual theory: it is a collective coordinate for the shape of the hyperplane. This is already evident from the fact that a constant gauge background (8.6.18) corresponds to a uniform translation of the hyperplane. An xµ -dependent background would by the same token go over to an xµ -dependent translation, a curved hyperplane, and the quanta of the ﬁeld A25 ii correspond to small oscillations of the hyperplane. This is the same phenomenon as with spacetime itself. We start with strings in a ﬂat background and discover that a massless closed string state corresponds to ﬂuctuations of the geometry. Here we found ﬁrst a ﬂat hyperplane, and then discovered that a certain open string state corresponds to ﬂuctuations of its shape. We should not be surprised that the hyperplane has become dynamical. String theory contains gravity. A gravitational wave passing through the hyperplane would warp spacetime itself, so the hyperplane could hardly remain rigid. Thus the hyperplane is indeed a dynamical object, a Dirichlet membrane, or D-brane for short. The p-dimensional D-brane, from dualizing 25 − p dimensions, is a Dp-brane. In this terminology, the original U(n) open string theory contains n D25-branes. A D25-brane ﬁlls space, so the string endpoint can be anywhere: it just corresponds to an ordinary Chan–Paton factor. Since T -duality interchanges Neumann and Dirichlet boundary conditions, a further T -duality in a direction tangent to a Dp-brane reduces it to a D(p − 1)-brane, while a T -duality in an orthogonal direction turns it into a D(p + 1)-brane. The case of a nontrivial angle will arise shortly. We could have started by investigating Dirichlet boundary conditions in their own right. The route we have taken, discovering them via T -duality, is a good way to develop many of their properties. It also shows that one can reach a state containing D-branes from an ‘ordinary’ state by a continuous process. That is, take the R → 0 limit of the original theory, leaving n D-branes in a noncompact space. In the superstring, we will use this to argue that D-branes are an essential part of the nonperturbative deﬁnition of the theory. For the bosonic string we will not try to make this argument because there is no strong argument that the theory has a nonperturbative deﬁnition. It is possible that bosonic string theory exists only as a mutilation of superstring theory. It is interesting to look at the U(n) symmetry breaking in the T -dual picture. When no D-branes coincide there is just one massless vector on each, giving gauge group U(1)n in all. If r D-branes coincide, there are new

270

8 Toroidal compactiﬁcation and T -duality

massless states because strings that are stretched between these branes can have vanishing length: the ∆X 25 in the mass formula (8.6.20) vanishes for n = 0 if both i and j are in this set of r. Thus, there are r2 vectors, forming the adjoint of a U(r) gauge group. This is just as found in the dual Wilson line picture. It is striking, however, that there will also be r2 massless scalars from the components normal to the D-brane. We will discuss the signiﬁcance of this below. The D-brane action The massless ﬁelds on the world-volume of a Dp-brane are a U(1) vector plus 25 − p world-brane scalars describing the ﬂuctuations. The low energy eﬀective action for this system is as always a very useful thing to consider. We now take the dual radius R to inﬁnity, focusing on one D-brane in the resulting 26-dimensional spacetime. The world-brane ﬁelds are in interaction with the massless closed string ﬁelds, whose action has already been discussed in section 3.7. Introduce coordinates ξ a , a = 0, . . . p on the brane. The ﬁelds on the brane are the embedding X µ (ξ) and the gauge ﬁeld Aa (ξ). We claim that the action is S p = −Tp

dp+1 ξ e−Φ [− det(Gab + Bab + 2πα Fab )]1/2 ,

(8.7.2)

with Tp a constant to be determined later. Here Gab (ξ) =

∂X µ ∂X ν Gµν (X(ξ)) , ∂ξ a ∂ξ b

Bab (ξ) =

∂X µ ∂X ν Bµν (X(ξ)) ∂ξ a ∂ξ b

(8.7.3)

are the induced metric and antisymmetric tensor on the brane. All features of the action (8.7.2) can be understood from general reasoning. Considering ﬁrst just the spacetime metric and the embedding, the simplest and lowest-derivative coordinate-invariant action is the integral of (− det Gab )1/2 , the world-volume. Note that this term has an implicit dependence on X µ (ξ) through the induced ﬁelds (8.7.3). Expanded around a ﬂat D-brane it gives the action for the ﬂuctuations, just as the Nambu– Goto action describes the ﬂuctuations of a string. The dilaton dependence e−Φ ∝ gc−1 arises because this is an open string tree-level action. The self-interactions of the open string ﬁelds, and their couplings to the closed string ﬁelds, arise ﬁrst from the disk. The dependence on Fab can be understood by using T -duality. Consider a D-brane that is extended in the X 1 - and X 2 -directions, with the other dimensions unspeciﬁed, and let there be a constant gauge ﬁeld F12 on it. Go to the gauge A2 = X 1 F12 . Now take the T -dual along the 2-direction. The Neumann condition in this direction is replaced by a Dirichlet condition, so the D-brane loses a dimension. However, the T -duality relation (8.6.18)

271

8.7 D-branes

between the potential and coordinate implies that the D-brane is tilted in the (1–2)-plane, X 2 = −2πα X 1 F12 .

(8.7.4)

The tilt gives a geometric factor in the action,

dX 1 1 + (∂1 X 2 )2

1/2

=

dX 1 1 + (2πα F12 )2

1/2

.

(8.7.5)

For any D-brane, by boosting to align it with the coordinate axes and then rotating to bring Fab to block-diagonal form, one can reduce the action to a product of factors (8.7.5) in each plane, equivalent to the Fab term in the determinant (8.7.2). This determinant form for the gauge ﬁeld action is known as the Born–Infeld action. It was originally proposed, unsuccessfully, as a solution to the short-distance divergences of quantum electrodynamics. Finally, the dependence on Bab is given by the following argument. The closed string ﬁeld Bµν and the open string ﬁeld Aµ appear in the string world-sheet action as3 i 2 1/2 ab µ ν d σ g ∂ X ∂ X B + i dX µ Aµ . (8.7.6) a µν b 4πα M ∂M Associated with each of these ﬁelds is a spacetime gauge invariance, which must be preserved for the consistency of the spacetime theory. The ordinary gauge transformation δAµ = ∂µ λ

(8.7.7)

is an invariance of the action (8.7.6), the boundary term changing by the integral of a total derivative. The antisymmetric tensor variation δBµν = ∂µ ζν − ∂ν ζµ

(8.7.8)

similarly changes the bulk action by a total derivative, but on a bounded world-sheet this gives rise to a surface term. This can only be canceled if the open string ﬁeld Aµ also transforms under the tensor gauge symmetry, δAµ = −ζµ /2πα .

(8.7.9)

Now, however, only the combination Bµν + 2πα Fµν ≡ 2πα Fµν 3

For the reader familiar with diﬀerential forms, this is i 2πα

B+i M

A. ∂M

The reader can translate the subsequent equations similarly.

(8.7.10)

272

8 Toroidal compactiﬁcation and T -duality

is invariant under both symmetries, and it is this combination that must appear in the action. Thus the form (8.7.2) of the action is fully determined. As a check, it is natural that the combination Gµν + Bµν should appear, since this same combination appears in the world-sheet action in conformal gauge. Incidentally, the action (8.7.2) can also be determined by taking the low energy limit of various open and open plus closed string amplitudes, but this is rather more laborious. For n separated D-branes, the action is n copies of the action for a single D-brane. We have seen, however, that when the D-branes are coincident there are n2 rather than n massless vectors and scalars on the brane, and we would like to write down the eﬀective action governing these. The ﬁelds X µ (ξ) and Aa (ξ) will now be n × n matrices. For the gauge ﬁeld the meaning is obvious — it becomes a non-Abelian U(n) gauge ﬁeld. For the collective coordinate X µ , however, the meaning is mysterious: the collective coordinates for the embedding of n D-branes in spacetime are now enlarged to n × n matrices. This ‘noncommutative geometry’ has proven to play a key role in the dynamics of D-branes, and there are conjectures that it is an important hint about the nature of spacetime. We can gain more insight by considering the eﬀective action. There is now a nonderivative term in the action, a potential for the collective coordinates, which can be deduced by T -duality from a constant Am ﬁeld. For such a vector potential, the ﬁeld strength is just [Am , An ], which becomes (2πα )−2 [Xm , Xn ] in the T -dual picture. Expanding in the ﬁeld strength, the leading term in the action is roughly V ∝ Tr ([Xm , Xn ][X m , X n ]) .

(8.7.11)

This potential has the property that at the point X m = 0 its second derivative vanishes, so that all kn2 scalars are massless; again k = 25 − p is the number of dualized dimensions. However, the space of ﬂat dimensions is smaller. The potential is the sum of squares and so vanishes only if all of the [X m , X n ] vanish. We can then use the U(n) gauge symmetry to go to a gauge in which the X m are all diagonal. There are thus kn ﬂat directions, just the number of diagonal elements. These kn diagonal elements can then be interpreted as the collective coordinates of the n Dp-branes. Thus the potential (8.7.11) correctly interpolates the physics of separated and coincident D-branes. When the X m do commute, the action should reduce to that of n separate D-branes, so we must have S p = −Tp

#

dp+1 ξ Tr e−Φ [− det(Gab + Bab + 2πα Fab )]1/2 $

+O([X m , X n ]2 ) .

(8.7.12)

The determinant is on the world-volume indices ab, while the trace is over

8.7 D-branes

273

the n Chan–Paton indices. The trace is the appropriate U(n) invariant which for diagonal matrices reduces to a sum over the separate D-branes. The full dependence on the commutator is more complicated than the simple potential (8.7.11), with additional couplings to other ﬁelds and corrections at higher orders in the commutator. However, the key property, the form of the ﬂat directions, is unaﬀected. The full dependence can be determined by starting from the Born–Infeld action for the fully Neumann case and performing the T -duality. Incidentally, higher derivative terms involving commutators of ﬁeld strengths cannot be determined by T duality alone. The D-brane tension It is instructive to compute the constant Tp , and for the superstring the precise value will be signiﬁcant. Before we carry out the explicit calculation, it is interesting to note that one gets a recursion relation for Tp from T duality. Note that in a constant dilaton background, the tension of a Dp-brane is Tp e−Φ : this is minus the action per unit volume for a static Dp-brane. Consider now a Dp-brane in which the p directions tangent to the D-brane have been periodically identiﬁed. That is, the Dp-brane is wrapped on a p-torus in spacetime. Its mass is its tension times the volume of the torus, Tp e−Φ

p

(2πRi ) .

(8.7.13)

i=1

Now take the T -dual on one of the periodic dimensions X p . This does not change the mass, being just a new description of the same state. In terms of the dilaton (8.3.31) of the T -dual theory, the mass (8.7.13) is 2πα1/2 Tp e−Φ

p−1

(2πRi ) .

(8.7.14)

i=1

In the dual theory, however, it is a D(p − 1)-brane wrapped on a (p − 1)torus, so its mass is Tp−1 e−Φ

p−1

(2πRi ) .

(8.7.15)

i=1

Equating the masses (8.7.14) and (8.7.15) gives Tp = Tp−1 /2πα1/2 , which determines the Tp up to one overall normalization.

(8.7.16)

274

8 Toroidal compactiﬁcation and T -duality

σ2 σ1

Fig. 8.5. Exchange of a closed string between two D-branes. Equivalently, a vacuum loop of an open string with one end on each D-brane.

To determine the actual value of the D-brane tension we need to calculate a string amplitude. For example, we could deduce it from the gravitational coupling to the D-brane, given by the disk with a graviton vertex operator. However, this involves the yet-unknown ratio gc /go2 , the closed string coupling arising from the vertex operator and the open string coupling from the disk. Instead we can obtain the absolute normalization as follows. Consider two parallel Dp-branes at positions X1m = 0 and X2m = y m . These two objects can feel each other’s presence by exchanging closed strings as shown in ﬁgure 8.5. This string amplitude is an annulus, with no vertex operators, and so can be calculated by the same method as in the previous chapter. The poles from graviton and dilaton exchange then give the coupling Tp of closed string states to the D-brane. Recall that in section 7.4 we calculated the annulus vacuum amplitude in terms of a loop of open strings, but then found closed string poles from the t → 0 limit of the modular integration. Here it is just those poles that we are interested in, but once again the simplest way to calculate the amplitude is to think of it as an open string loop. In fact, the earlier result (7.4.1) carries over with just a few changes. The number of momentum integrations is reduced from 26 to p + 1 and similarly V26 becomes Vp+1 ; the weight hi acquires an additional term y 2 /4π 2 α from the tension of the stretched open string; the Chan–Paton weight n2 becomes 2 (since the string can attach with either orientation). Thus we

275

8.7 D-branes have A = iVp+1 =

∞ dt

t

0

(8π 2 α t)−(p+1)/2 exp(−ty 2 /2πα )η(it)−24

iVp+1 2 (8π α )(p+1)/2

∞

dt t(21−p)/2 exp(−ty 2 /2πα )

0

×[exp(2π/t) + 24 + . . . ] ,

(8.7.17)

where the asymptotics have been obtained just as in section 7.4. The ﬁrst term is from tachyon exchange and so is again an uninteresting bosonic artifact. The integral over the second term gives

24 23 − p A = iVp+1 12 (4π 2 α )11−p π (p−23)/2 Γ |y|p−23 2 2 24π = iVp+1 10 (4π 2 α )11−p G25−p (y) , (8.7.18) 2 where Gd (y) is the massless scalar Green’s function in d dimensions, the inverse of −∇2 . We now need to compare this with a ﬁeld theory calculation of the same amplitude. Since the antisymmetric tensor does not couple to the D-branes, the relevant terms from the spacetime action (3.7.25) are 1 S= 2 2κ

˜ µΦ ˜ − 1 ∇µ Φ ˜∇ ˜ ˜ 1/2 R d X (−G) 6 26

.

(8.7.19)

Recall that the tilde denotes the Einstein metric; this is the convenient form to use because its action decouples from that of the dilaton. The tilded dilaton has been shifted so that its expectation value is zero. In terms of the same variables, the relevant terms from the D-brane action (8.7.2) are p − 11 ˜ ˜ ab )1/2 . Φ (− det G (8.7.20) S p = −τp dp+1 ξ exp 12 We have deﬁned τp = Tp e−Φ0 ; this is the actual physical tension of the Dp-brane when the background value of the dilaton is Φ0 . The ﬁeld theory graph analogous to ﬁgure 8.5 is the exchange of a single graviton or dilaton between the D-branes. To obtain the propagator we ˜ µν − ηµν and to expand the spacetime action to second order in hµν = G ˜ Also, we need to choose a gauge for the gravitational second order in Φ. calculation. The simplest gauge for perturbative calculation is 1 (8.7.21) Fν ≡ ∂µˆ hµν − ∂ν hµˆµ = 0 , 2 where a hat indicates that an index has been raised with the ﬂat space metric η µν . Expanding the action to second order and adding a gauge-

276

8 Toroidal compactiﬁcation and T -duality

ﬁxing term −Fν F νˆ /4κ2 , the spacetime action becomes

1 2 ˜ µˆ ˜ 1 ˆ ˆ Φ . (8.7.22) d26 X ∂µ hνλ ∂µˆ hνˆ λ − ∂µ hνˆν ∂µˆ hλλ + ∂µ Φ∂ 2 8κ 2 3 Inverting the kinetic term, one obtains the momentum space propagators S=−

(D − 2)iκ2 , (8.7.23a) 4k 2 2iκ2 2

hµν hσρ = − 2 ηµσ ηνρ + ηµρ ηνσ − ηµν ησρ . (8.7.23b) k D−2 For later reference we have given these for general D. The D-brane action expanded around a ﬂat conﬁguration is p − 11 ˜ 1 p+1 Φ − haa . (8.7.24) S p = −τp d ξ 12 2 Note that hµν here is traced only over the directions tangent to the D-brane; we have taken ξ to be rectilinear coordinates with metric δab . We can now read oﬀ the Feynman graph, ˜ Φ

˜ =−

Φ

iκ2 τ2p p − 11 A = 2 Vp+1 6 12 k⊥ =

6iκ2 τ2p Vp+1 . 2 k⊥

2

1 1 + 2(p + 1) − (p + 1)2 2 12

(8.7.25)

Comparison with the string result (8.7.18) yields π τ2p = (4π 2 α )11−p . (8.7.26) 256κ2 It is a useful check that, once the dust has cleared, the recursion relation from T -duality is satisﬁed. As one application, consider a state with n 25-branes, which is the same as an ordinary (fully Neumann) n-valued Chan–Paton factor. Expanding the 25-brane action (8.7.12) to second order in the gauge ﬁeld gives τ25 (8.7.27) (2πα )2 Tr(Fµν F µν ) . 4 This relates the open string gauge coupling to the gravitational coupling. Using eqs. (6.5.14), (6.5.16), (6.6.15), and (6.6.18), we can write it as a relation between the vertex operator normalizations gc and go , 4πα go2 go2 = 218 π 25/2 α6 . = (8.7.28) gc κ This has the right form, the square of the open string coupling being proportional to the closed string coupling, but now with the numerical coeﬃcient included. We have essentially obtained this via unitarity, factoring on the closed string poles of the annulus diagram. We could also have

8.8 T -duality of unoriented theories

277

done this more directly, as in exercise 7.9, without the excursion through D-branes. 8.8

T -duality of unoriented theories

The R → 0 limit in an unoriented string theory also leads to interesting new physics. Consider closed strings ﬁrst. To form the unoriented theory we impose Ω = +1 on the states. Following the discussion in section 8.5, we can think about this as gauging Ω. In particular, the transition functions used to build the world-sheet may now include orientation reversal, since it is a gauge symmetry, and this produces unoriented world-sheets. The T -dual theory is obtained by using the coordinate X m (z, ¯z ) = m XL (z) − XRm (¯z ) in place of X m (z, ¯z ) = XLm (z) + XRm (¯z ). In the original description we are gauging world-sheet parity Ω, which acts as Ω : XLM (z) ↔ XRM (z) .

(8.8.1)

In terms of the T -dual coordinates, this is Ω : X m (z, ¯z ) ↔ −X m (¯z , z) , X µ (z, ¯z ) ↔ X µ (¯z , z) ,

(8.8.2a) (8.8.2b)

where again m indexes the coordinates on which the T -dual has been taken and µ those on which it has not. In the dual picture, the symmetry (8.8.2) is thus the product of a world-sheet parity transformation and a spacetime reﬂection. We have seen that gauging world-sheet parity alone produces the unoriented theory, while gauging the reﬂection alone produces an orbifold. The result of the combined projection is known as an orientifold. Orientifolds are rather similar to orbifolds. Separate the string wavefunction into its internal part and its dependence on the center of mass xm , and take the internal wavefunction to be an eigenstate of Ω. The projection onto Ω = +1 then determines the string wavefunction at −xm to be the same as that at xm , up to a sign. For example, the various components of the metric and antisymmetric tensor satisfy Gµν (x ) = Gµν (x) , Bµν (x ) = −Bµν (x) , Gµn (x ) = −Gµn (x) , Bµn (x ) = Bµn (x) , Gmn (x ) = Gmn (x) , Bmn (x ) = −Bmn (x) ,

(8.8.3a) (8.8.3b) (8.8.3c)

where (xµ , xm ) = (xµ , −xm ). There is a minus sign for each m, n and an additional minus sign for each BMN ; the orbifold is the same except that the latter sign would be absent. In other words, the T -dual spacetime is the torus T k identiﬁed under a Z2 reﬂection in the k compact directions, just as in the orbifold construction. In the case of a single periodic dimension,

278

8 Toroidal compactiﬁcation and T -duality

(a)

(b)

Fig. 8.6. Identiﬁcations deﬁning (a) the torus and (b) the Klein bottle.

for example, the dual spacetime is the line segment 0 ≤ x25 ≤ πR , with orientifold ﬁxed planes at the ends. Notice also that away from the orientifold ﬁxed planes, the local physics is that of the oriented string theory. Unlike the original unoriented theory, where the projection removes half the states locally, here it relates the string wavefunction at any point to its value at the image point, as in eq. (8.8.3). One diﬀerence between the orbifold and orientifold constructions is that in the latter there is no direct analog of the twisted states, because the Klein bottle does not have the modular transformation τ → −1/τ. Consider ﬁgure 8.6, which shows the torus and Klein bottle. On the torus, the projection operator inserts a twist in the timelike direction of ﬁgure 8.6(a); rotating the ﬁgure by 90◦ , this becomes a twist in the spacelike direction, implying twisted states in the spectrum. However, if the Klein bottle of ﬁgure 8.6(b) is rotated by 90◦ , the directions of time on the two opposite edges do not match and there is no interpretation in terms of intermediate states in this channel. It should be noted then that orientifold planes cannot be dynamical. Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent ﬂuctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identiﬁcations (8.8.3) become boundary conditions at the ﬁxed plane, such that the incident and reﬂected waves cancel. For the D-brane, the reﬂected wave is higher order in the string coupling. We draw oriented strings with arrows to indicate the orientation. In the unoriented theory, we can either omit the arrow or take a linear combination of the two orientations. The latter picture is more consistent with the idea of gauging the discrete symmetry, and is more general: in an orientifold the parity operation is accompanied by a spacetime transformation, so we cannot just forget the arrow. Although we have introduced the orientifold construction via T -duality, one can consider more general orientifolds that are not simply T -duals

279

8.8 T -duality of unoriented theories

-θ2 R'

-θ1R'

0

θ 1R'

θ 2 R'

π R'

Fig. 8.7. Orientifold planes at 0 and πR , D-branes at θ1 R and θ2 R , and Dbrane images at −θ1 R and −θ2 R . The twist operator Ω acts on any string by a combination of a spacetime reﬂection and reversing the orientation arrow, so that the string running from 2 to the image of 1 becomes a string running from 1 to the image of 2.

of toroidal compactiﬁcations, gauging a group that combines world-sheet parity with various spacetime symmetries. Open strings In the case of open strings, the situation is similar. Let us focus for convenience on a single compact dimension. Again there is one orientifold ﬁxed plane at 0 and another at πR . Introducing S O(n) Chan–Paton factors (the symplectic case works in the same way), a general Wilson line can be brought to diagonal form, and for n even the eigenvalues are paired:

W = diag eiθ1 , e−iθ1 , eiθ2 , e−iθ2 , · · · , eiθn/2 , e−iθn/2 .

(8.8.4)

Thus in the dual picture there are 12 n D-branes on the line segment 0 ≤ X 25 ≤ πR , and 12 n at their image points under the orientifold identiﬁcation. Strings can stretch between D-branes and their images as shown in ﬁgure 8.7. The generic gauge group is U(1)n/2 . As in the oriented case, if r D-branes are coincident there is a U(r) gauge group. However, if now the r D-branes in addition lie at one of the ﬁxed planes, then strings stretching between one of these branes and one of the image branes also become massless and we have the right spectrum of additional states

280

8 Toroidal compactiﬁcation and T -duality

to ﬁll out S O(2r). The maximal S O(n) is restored if all of the branes are coincident at a single orientifold plane. Note that this maximally symmetric case is asymmetric between the two ﬁxed planes. If n is odd, the ﬁnal eigenvalue of W is ±1, so that in the T -dual picture it is ﬁxed to lie on one or the other orientifold plane. Having no image, it is actually a half D-brane, as measured for example by its mass. Also, if we consider for n = 2 the Wilson line diag(1, −1) (which is in O(2) but not S O(2)), then rather than one D-brane and its image there are two half D-branes, one at each ﬁxed plane. The orientifold plane, like the D-brane, couples to the dilaton and metric. The amplitude is just as in the previous section, with the Klein bottle and M¨ obius strip replacing the annulus. In fact, we have already done the relevant calculation in chapter 7. There we found that the total dilaton coupling cancels for S O(213 ). In the T -dual picture, the total dilaton coupling of the orientifold planes cancels that of 212 D-branes (the images don’t count!). If we T -dualize on k = 25 − p dimensions there are 2k ﬁxed planes, where the coordinates are all combinations of 0 and πRm . Thus the eﬀective action for a single ﬁxed plane is 212−k Tp

dp+1 ξ e−Φ (− det Gab )1/2 ,

(8.8.5)

the integral running over the ﬁxed plane. Although the orientifold has no direct analog of twisted sectors, it has been argued that there is a sense in which the twisted states are open strings. We prefer not to emphasize this, as the analogy is not complete: gauging Ω does not of itself introduce world-sheet boundaries. However, there is a deﬁnite physical sense in which it is natural to add open strings to an orientifold theory, to cancel the dilaton tadpole. This cancellation will play an important role in the superstring. Exercises 8.1 (a) Derive the equal time commutator (8.2.21). (b) Work out the details showing that vertex operators of the form (8.2.22) commute at equal time and ﬁnite separation. 8.2 For a single periodic dimension, calculate the scattering amplitude for four closed string tachyons with winding numbers +1, −1, 0, and 0, and vanishing compact momenta. 8.3 We have shown T -duality for spacetime-independent backgrounds. We would expect that a background with a varying metric G25,25 (xµ ) would be T -dual to one with the inverse metric G25,25 (xµ )−1 . In principle we can show this by expanding around the static conﬁguration, but it is useful to show it directly as follows.

Exercises

281

(a) Write the world-sheet action for a string moving in a general massless background, with only the requirement that the background be invariant under translations of X 25 . Let the coordinate X 25 have periodicity R. (b) Show that replacing ∂a X 25 with ∂a X 25 + Aa makes the X 25 translations a local symmetry on the world-sheet. (c) Add a term iφFab to the world-sheet action, with φ(σ) a new ﬁeld and Fab the ﬁeld strength for Aa . The path integral over φ forces the ﬁeld strength to vanish, so the gauge ﬁeld is trivial: this eﬀectively undoes step (b). Argue that in the gauge ∂a Aa = 0 the resulting theory is just a rewriting of the original one. (d) Show that you can reach the gauge X 25 = 0, and then carry out the Gaussian path integration over Aa (treat it classically: ignore the determinant). Show that the resulting action has the expected form for the T -dual theory, if φ is identiﬁed with X 25 times an appropriate constant. To get the transformation (8.3.31) of the dilaton one would need to carry the Aa path integration to one loop. Also, to get the coordinate periodicity of X 25 requires a somewhat careful argument. (References: Buscher (1987, 1988), Roˇcek & Verlinde (1992).) 8.4 Verify the equality of the string partition function for several compact dimensions as calculated directly and via the path integral; see the comment below eq. (8.4.12). 8.5 (a) Show that the spectrum (8.4.26) for a single compact dimension deﬁnes an even self-dual lattice. (b) Do the same for the spectrum (8.4.7). 8.6 In the example of two periodic dimensions, section 8.4, show that the general kinetic term (8.4.35) reduces to the form (8.4.39). Verify the assertions about the action of the various dualities on τ and ρ in the special case that τ and ρ are both imaginary. 8.7 (a) Calculate the scattering amplitude for ripples on a Dp-brane. That is, consider the tree-level amplitude with four collective coordinate vertex operators. Note that the polarizations are orthogonal to all momenta, a major simpliﬁcation. (b) Examine the Regge and hard scattering limits. (c) Take the low momentum limit and compare with what is obtained from the action (8.7.2). 8.8 Repeat the previous problem for two coincident Dp-branes. The collective coordinates now include Chan–Paton factors. Note in particular the change in part (c). 8.9 (a) Calculate the expectation value of n exponential vertex operators in the interior of the disk, with Neumann boundary conditions on p + 1 coordinates and Dirichlet conditions on 25 − p.

282

8 Toroidal compactiﬁcation and T -duality

(b) Extend the calculation to include operators ∂a X M in the interior. 8.10 From the result of the previous exercise, calculate the amplitude for a Dp-brane to emit a graviton. Compare with the Dp-brane eﬀective action and the result (8.7.28) for the couplings. 8.11 (a) For the boundary conditions of exercise 8.9, calculate the disk amplitude with two closed string tachyons: this corresponds to scattering a closed string from a D-brane. (b) Examine the Regge and hard scattering limits. (c) Show that the amplitude has poles from intermediate open strings. Give a physical interpretation. 8.12 (a) Do the calculation of the previous problem for the messier but more interesting case of massless closed strings. (b) Show that there are poles in the graviton amplitude due to the open string collective coordinate, and compare with a calculation using the action (8.7.2). (c) Do the same for the antisymmetric tensor and the open string photon.

9 Higher order amplitudes

In this chapter we discuss higher order amplitudes in string theory. The main goal of the ﬁrst half of the chapter is an understanding of the ﬁniteness and unitarity of string perturbation theory. After discussing the tree-level amplitude in some detail, we consider general amplitudes, using the idea of factorization to relate higher order amplitudes to lower order ones. In the second half, we brieﬂy discuss some ideas that have been put forward to understand all-order and even nonperturbative string theory. The more powerful methods based on supersymmetry will not be developed until volume two. The present chapter is outside the main line of development in this book and can be skipped on a ﬁrst reading. 9.1

General tree-level amplitudes

Chapter 6 was primarily concerned with the calculation of some speciﬁc tree-level amplitudes. It is useful to begin the present chapter with an analysis of the general tree amplitude, as it illustrates some patterns that will arise in the higher order amplitudes. We focus on the closed oriented string, and on the three- and four-point amplitudes, which illustrate the important features. The main issues are convergence and ‘unitarity.’ The last term is used somewhat loosely to cover both unitarity of the S-matrix, S S † = 1, and positivity of the Hilbert space. Of course we already know from chapter 4 that the BRST cohomology has a positive inner product, so what we need to show is that amplitudes respect the BRST invariance: that only BRST-invariant states are produced, and that the amplitudes for BRST-equivalent states are equal. We have given a general argument for the latter in section 5.4, but it is useful to illustrate it explicitly. Before going on to a general analysis let us record some results that are useful in calculation. Suppose we have d inﬁnite ﬂat dimensions and the rest compact (or replaced by a compact unitary CFT), and the tachyon 283

284

9 Higher order amplitudes

vertex operator is normalized gc,d : eik·X : .

(9.1.1)

For example, with toroidal compactiﬁcation of one dimension we would have gc,25 = gc (2πR)−1/2 as in the previous chapter. Then for the matter part of a general OCQ vertex operator, the correct normalization can be ﬁxed by the most singular term in the OPE 2 gc,d + ... . (9.1.2) z 2 ¯z 2 A bar denotes the Euclidean adjoint (6.7.28). The singularity is |z|−4 because the vertex operator is of weight (1, 1). A repetition of the tachyon pole calculation (6.6.7) then determines the normalization

Vj (k; z, ¯z )Vj (k; 0, 0) =

e−2λ ˜c(¯z1 )c(z1 ) ˜c(¯z2 )c(z2 ) ˜c(¯z3 )c(z3 ) : eik·X (z, ¯z ) : =

8πi |z12 z13 z23 |2 (2π)d δ d (k) . 2 α gc,d

(9.1.3)

So all normalization constants in the S-matrix are known in terms of gc,d = κd /2π. Three-point amplitudes Let us ﬁrst express the tree amplitude in several useful forms. The gaugeﬁxed expression (5.3.9) for the string S-matrix reduces, for the general tree-level, three-point amplitude, to a path integral with three ﬁxed vertex operators:

ˆ (∞, ∞)V ˆ (1, 1)V ˆ (0, 0) SS2 (1; 2; 3) = e−2λ V 2 3 1

S2

,

(9.1.4)

where each numeral ‘n’ is an abbreviation for string momentum and state, (kn , jn ). A ˆ is used to designate a vertex operator in the ‘ﬁxed’ picture — that is, not including the b-ghost insertions for the integral over position. As we have discussed in section 5.4, this is the form that comes naturally out of the state–operator mapping, and in the OCQ case takes the form ˆ j = ˜ccVj . (9.1.5) V Using the results from section 6.7 for translating between path integrals on the sphere and matrix elements, this amplitude can also be represented as a matrix element or as an operator product coeﬃcient: ˆ 1 |V ˆ 2 (1, 1)|V ˆ 3 = e−2λ c123 . SS (1; 2; 3) = e−2λ

V (9.1.6) 2

This relates the three-point amplitude in spacetime to a matrix element on the world-sheet, and further to the operator product coeﬃcients which encode the local properties of the CFT.

9.1 General tree-level amplitudes

285

The ﬁrst issue is the decoupling of BRST-null states, so that BRSTˆ 3 in equivalent states have equal amplitudes. Suppose that the state |V the three-point amplitude is null, ˆ 3 = QB |χ . |V (9.1.7) Then ˆ 2 (1, 1)|χ

ˆ 1 | [V ˆ 2 (1, 1), QB ] |χ +

V ˆ 1 |QB V e2λ SS2 (1; 2; 3) =

V =0, (9.1.8) the terms vanishing respectively by the BRST invariance of states 2 and ˆ 2 = {QB , χ} or |V ˆ 1 = QB |χ the amplitude also vanishes. 1. Similarly, if V We have used the Hilbert space formalism here, but it is worthwhile to phrase this same argument in path integral language. Suppose V3 is null, ˆ 3 = QB · X . V (9.1.9) Expand the contour integral on the sphere away from X until it becomes ˆ 1 plus a small loop around V ˆ 2 . These then vanish a small loop around V ˆ ˆ by BRST invariance, QB · V1 = QB · V2 = 0. To this order then, amplitudes are well deﬁned on the physical Hilbert space, the BRST cohomology. The decoupling of the null graviton, associated with spacetime coordinate invariance, is a special case of the decoupling of BRST-null states. It is then natural to wonder whether spacetime coordinate invariance is a piece of some much larger symmetry in string theory. We will discuss some ideas in this direction in section 9.6. We now consider the unitarity of the S-matrix,

∗ Smn Spn = δmp , ?

(9.1.10)

n

which is equivalent to the conservation of probability. It is useful to break up the S-matrix into a term in which nothing happens and a scattering term, Smn = δmn + iTmn .

(9.1.11)

For the scattering matrix T , unitarity implies ∗ Tmp − Tpm =i

∗ Tmn Tpn .

(9.1.12)

n

The T-matrix by deﬁnition involves a total of at least three particles, so for three-particle amplitudes on the sphere the right-hand side of eq. (9.1.12) vanishes. Unitarity then requires TS2 (1; 2; 3) = TS2 (−1; −2; −3)∗ ,

(9.1.13)

where a minus sign is shorthand for replacing an incoming vertex operator by an outgoing vertex operator for the same state, or vice versa. The

286

9 Higher order amplitudes

condition (9.1.13) is satisﬁed if the incoming and outgoing vertex operators are related ˆ −i = −V ˆi. V

(9.1.14)

This is a natural deﬁnition: conjugation takes eik·X into e−ik·X , turning a vertex operator that injects momentum k µ into the world-sheet into one that absorbs it, and similarly for other quantum numbers. The minus sign is included to undo the reversal of the order of ˜cc by the adjoint, so that ˆ −i is obtained simply by conjugating all explicit factors of i in the vertex V operators. Complex conjugation of the path integral then gives

ˆ (∞, ∞)V ˆ 2 (1, 1)V ˆ 3 (0, 0) V 1

∗

ˆ (∞, ∞)V ˆ −2 (1, 1)V ˆ −3 (0, 0) =− V , −1 S2 S2 (9.1.15) implying the unitarity relation (9.1.13). Naively, the path integral is manifestly real and so the relation (9.1.15) should hold without the minus sign. However, we must remember that the path integral has to be deﬁned by speciﬁc analytic continuations, and the explicit result (6.2.6) acquired an overall factor of i in this way. Four-point amplitudes and world-sheet duality The connected four-point amplitude is SS2 (1; 2; 3; 4) = e−2λ

C

ˆ (∞, ∞) V ˆ 2 (1, 1) V ˆ 3 (0, 0) V4 (z, ¯z ) d2 z V 1

S2

.

(9.1.16)

Writing the expectation value as a matrix element gives ˆ 1 |T[ V ˆ 2 (1, 1) V4 (z, ¯z ) ]|V ˆ 3 .

V

(9.1.17)

The ghost expectation value is just a constant for OCQ vertex operators, but to make the connection with the more general case it is useful to move the ghosts around a bit. Using a contour argument one can write the ghost insertion for the d2 z integration as ˆ 4 (z, ¯z ) ] } . V4 (z, ¯z ) = (z¯z )−1 {b0 , [ ˜b0 , V

(9.1.18)

Using this and the condition (4.3.29) that b0 and ˜b0 annihilate the physical states, the matrix element becomes ˜

ˆ 4 (1, 1)|V ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 z L0 −1 ¯z L0 −1 V ˆ 3

θ(1 − |z|)

V ˜ ˆ 2 (1, 1)|V ˆ 1 |V ˆ 4 (1, 1)b0 ˜b0 z −L0 −1 ¯z −L0 −1 V ˆ 3 . (9.1.19) + θ(|z| − 1)

V Let us also write the matrix element after insertion of a complete set of

9.1 General tree-level amplitudes

287

intermediate states, θ(1 − |z|)

i,i

+ θ(|z| − 1)

i,i

2

2

2

2

z α (ki +mi )/4−1 ¯z α (ki +m˜ i )/4−1

ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 4 (1, 1)|V ˆ 3

×

V

z −α (ki +mi )/4−1 ¯z −α (ki +m˜ i )/4−1 2

2

2

2

ˆ 1 |V ˆ 4 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 2 (1, 1)|V ˆ 3 . (9.1.20) ×

V

In this last form the behavior of the integral at z → 0 is evident, and it is the same as for the tachyon amplitudes studied in chapter 6. If ki2 + m2i ˜ 2i are positive for all i, that is, and ki2 + m 4 , (9.1.21) α then the integral converges. Elsewhere it is deﬁned as in eq. (6.6.7), ki2 = (k1 + k2 )2 >

|z|<1

d2 z z α (k

2 +m2 )/4−1

¯z α (k

2 +m ˜ 2 )/4−1

=

δm2 ,m˜ 2 8π . 2 α k + m2 − i

(9.1.22)

One can think of this as analytic continuation from large positive values of m2 , keeping m2 slightly below the real axis. Similarly, the integral as z → ∞ converges for (k1 + k4 )2 > 4/α and is deﬁned elsewhere by analytic continuation. The form (9.1.20) is not useful for studying the z → 1 limit. Instead use the symmetry of the amplitude to interchange vertex operators 2 and 3. The integral for z → 1 is then deﬁned by analytic continuation from the convergent region (k1 + k3 )2 > 4/α . The need to decompose the range of integration and to make possibly diﬀerent analytic continuations for the diﬀerent regions poses no diﬃculty. After the expansion (9.1.20) is integrated as in (9.1.22), the amplitude is expressed as a series of poles in s and u. We know from chapter 6 that the amplitude also has poles in t, but these are not explicit: they come from the divergences of the inﬁnite sums. Similarly the open string amplitude I(s, t) has poles in s and t, but can be written as an inﬁnite sum of terms having poles only in s, or only in t. This property is known as duality, or world-sheet duality to distinguish it from T - and other dualities. We can think of this as arising because the world-sheet can be cut open on diﬀerent curves and so written as sums over intermediate states in diﬀerent ways. Another example is the sum over open string states giving closed string poles on the cylinder. Unitarity of the four-point amplitude Now let us check that four-point amplitudes are equal for equivalent ˆ 3 = QB |χ be null. Using the form (9.1.19), we can commute states. Let |V

288

9 Higher order amplitudes

ˆ 1 |. The BRST operator commutes QB to the left until it annihilates

V ˜ with L0 , L0 and with the vertex operators, but not with b0 and ˜b0 . The nonzero commutators involve

˜ ˜ ˜ b0 ˜b0 z ±L0 −1 ¯z ±L0 −1 , QB = ±b0 ∂¯z z ±L0 −1 ¯z ±L0 ∓ ˜b0 ∂z z ±L0 ¯z ±L0 −1 . (9.1.23) These are total derivatives, illustrating the general principle (5.4.6) that the couplings of null states are proportional to total derivatives on moduli space. We have already seen an example of this for the massless vertex operators in section 6.6. As in that example, the integral vanishes by the canceled propagator argument. The integral at z → 0 is deﬁned by analytic continuation from large positive (k1 + k2 )2 . The surface term vanishes identically there, and so by analyticity must vanish everywhere. The same argument applies to the other limits of moduli space. For the four-particle amplitude on the sphere, the unitarity condition is TS2 (1; 2; 3; 4) − TS2 (−1; −2; −3; −4)∗ =i

j5

= −ie−4λ

dd−1 k5 TS (1; 2; 5)TS2 (−5; 3; 4) 2E5 (2π)d−1 2 + 2 permutations

j5

dd−1 k5 ˆ 1 |V ˆ 2 (1, 1)|V ˆ 5

V ˆ −5 |V ˆ 3 (1, 1)|V ˆ 4

V 2E5 (2π)d−1 + 2 permutations . (9.1.24)

The intermediate particle is on-shell, the sum j5 is over a complete set of physical states, and the right-hand side is summed over all ways to break 1234 into pairs: 12–34, 13–24, and 14–23. Eq. (9.1.24) is what we wish to show. Using the reality condition (9.1.13) in the matrix element (9.1.20), the left-hand side of eq. (9.1.24) vanishes in the region where the z integral converges. The nonzero part comes from the i introduced in the analytic continuation, k2

1 1 − 2 = 2πiδ(k 2 + m2 ) . 2 + m − i k + m2 + i

(9.1.25)

Thus, TS2 (1; 2; 3; 4) − TS2 (−1; −2; −3; −4)∗ 16π 2 i ˆ 1 |V ˆ 2 (1, 1)b0 ˜b0 |i Gii

i |V ˆ 3 (1, 1)|V ˆ 4

= 2λ δm2 ,m˜ 2 δ(ki2 + m2i )

V i i α e i,i + 2 permutations . (9.1.26) The resemblance to the desired unitarity relation (9.1.24) is obvious, and a little bookkeeping and deﬁnition-chasing established the result. The index i in the sum (9.1.26), running over a complete set of string states, labels

9.1 General tree-level amplitudes

289

both internal state and momentum, i ↔ (j, k) .

(9.1.27)

ˆ The delta functions and b0 ˜b0 factor in eq. (9.1.26) project onto the space H ˜ 0 , b0 , and ˜ b0 , which was introduced in the discussion annihilated by L0 , L of the BRST cohomology. Thus, i,i

δm2 ,m˜ 2 δ(ki2 + m2i )b0 ˜b0 |i Gii

i | i

i

= δL0 ,L˜ 0 δ(4L0 /α )b0 ˜b0 =−

iα 16π 2 e2λ

dd−1 k ˆ j (k)| . ˆ j (k)

V |V d−1 2E(2π) ˆ j∈H

(9.1.28)

k 0 =±ωk

The normalization of the basis |Vj (k) follows from the OPE (9.1.2) and the expectation value (9.1.3), which imply 2λ ˆ j (k)|˜c0 c0 |V ˆ j (k ) = 8πie δjj (2π)d δ d (k − k ) .

V (9.1.29) α Using the relation (9.1.28), the form and normalization of eq. (9.1.26) agree with the unitarity condition (9.1.24), leaving one important loose end. The sum in the unitarity relation runs only over the physical Hilbert space, the BRST cohomology, while the ﬁnal sum in (9.1.28) runs over a ˆ We now show that these sums are equal. Consider complete basis for H. ˆ First, take a complete set of ‘unphysical states’ the following basis for H. |a U . That is, no linear combination of these is annihilated by QB . Second, take a complete set of BRST-invariant states |b P which are independent in the sense that no linear combination of them is null. Finally, take the complete set of BRST-null states |a N = QB |a U to complete the basis. Deﬁne the operators P and U:

P|b P = |b P , P|a U = 0 , P|a N = 0 ; U|b P = 0 , U|a U = 0 , U|a N = |a U .

(9.1.30a) (9.1.30b)

Then 1 = P + QB U + UQB .

(9.1.31)

Insert this decomposition into the factorization relation. The ﬁrst term projects onto one state from each cohomology class, the physical spectrum appearing in the unitarity sum. The second and third terms each contain QB , and so vanish by the earlier argument that null states decouple. Thus unitarity is established. It is worth noting that the argument just given is very similar to the argument that amplitudes in gauge theory are simultaneously Lorentz-

290

9 Higher order amplitudes

invariant and unitary. The numerator of the covariant photon propagator η µν , (9.1.32) k2 has a negative eigenvalue, corresponding to a negative norm intermediate state. Choose any null vector nµ such that n · k = 1, and decompose −i

η µν = (η µν − k µ nν − nµ k ν ) + k µ nν + nµ k ν .

(9.1.33)

At k 2 = 0, the ﬁrst term has d − 2 positive eigenvalues and no negative eigenvalues. If the photon is coupled to conserved currents at both ends, the last two terms vanish, so the unitarity relation is satisﬁed with just the d − 2 positive norm states expected for a photon in d dimensions. The decomposition (9.1.31) is the generalization of eq. (9.1.33), with k µ → |a N and nµ → |a U . To conclude, let us review how various general principles are illustrated by this example. BRST null states decouple, but only after integration on moduli space. The potential surface terms from the latter vanish by the canceled propagator argument. The ghost manipulations leading to eq. (9.1.19) have a natural interpretation in terms of the Riemann surface picture from chapter 5. Consider the sphere with vertex operators at 0, 1, z, ∞. To be speciﬁc take |z| > 1 as in the second term of eq. (9.1.19). We can think of this sphere as being formed from two disks, with coordinates z1 and z2 . Put vertex operators at z1 = 0, 1 and at z2 = 0, 1, and join the disks by the identiﬁcation z1 = z/z2 . For the modulus z, we have

z1 ∂z1 = . ∂z z2 z

(9.1.34)

The ghost insertion (5.4.15) for dz is then b0 /z, and similarly for d¯z , as illustrated by eq. (9.1.19). 9.2

Higher genus Riemann surfaces

There are many diﬀerent representations of higher genus Riemann surfaces. The one that will be most useful for our purposes is based on the idea of building higher genus surfaces by sewing together lower genus surfaces. This will be described in the next section, but ﬁrst for completeness we describe some other common constructions. The treatment is necessarily rather brief. Schottky groups Imagine removing two disks from a sphere and joining the edges of the resulting holes. This produces a handle, so the surface obtained in this

9.2 Higher genus Riemann surfaces

291

way is a torus. Repeating g times, with a total of 2g holes, produces a genus g surface. We can make this more precise as follows. Regarding the sphere as the complex z-plane plus point at inﬁnity, consider the mapping z − U z−U =K , (9.2.1) z −V z−V which depends on three complex parameters U, V , and K. This is a M¨ obius transformation, and in particular is holomorphic and one-to-one. The points U and V are ﬁxed under γ. Taking |K| > 1, the surface expands away from U and contracts toward V . A fundamental region for γ is the annular region γ:

|K|

−1/2

z − U < |K|1/2 . < z−V

(9.2.2)

Identifying points under γ glues together the inner and outer boundaries of the annulus (9.2.2), producing a torus. The group generated by g such elements (9.2.1) produces the genus g surface, with restrictions on the parameters so that the sewn circles do not overlap. Every closed oriented Riemann surface can be obtained in this way. The range of the parameters is not simple, in part because some of the modular transformations act in a complicated way. Riemann surfaces with boundary and unoriented Riemann surfaces can be constructed by dividing by one or two reﬂections of a closed oriented surface, as we have illustrated in the lower genus examples. Let us see that the Schottky construction produces the expected number of moduli. Each transformation (9.2.1) has three complex parameters for a total of 3g. One can think of the six real parameters for each handle as being two for the position of each hole, one for the product of the radii of the holes (only the product, because if one hole is made smaller and the other larger we get the same surface), and one for the relative angle at which the holes are sewn. There is redundancy, because an overall M¨ obius transformation of the sphere gives an equivalent surface. Subtracting the six real parameters for this group leaves 6g − 6, the correct number of moduli for genus g ≥ 2 surfaces. For g = 1, two M¨ obius transformations leave the mapping invariant, so two real parameters remain. The Schottky parameterization shows that the moduli space has natural complex coordinates. The path integral has certain holomorphicity properties with respect to these coordinates, though we will not emphasize this. Having constructed the general surface as a coset of the sphere, it is evident that Green’s functions can be constructed by use of the method of images, summing over all images of a source. Functional determinants and other objects of interest can be expressed similarly.

292

9 Higher order amplitudes

B2

B1

A1

A2

B3

A3

Fig. 9.1. Genus-three surface with A- and B-cycles marked.

Fuchsian groups For this construction, consider ﬁrst the world-sheet metric. It is a theorem that by a Weyl transformation one can always make the world-sheet curvature R a constant. At genus zero this gives the round sphere, at genus one the ﬂat torus, and at higher genus a surface of constant negative curvature. The simply connected covering space for any of these will similarly have constant curvature. There are precisely three such simply connected spaces — the round sphere again, the ﬂat plane (the covering space of the torus) and the upper half-plane with metric ds2 =

dzd¯z . (Im z)2

(9.2.3)

The last is the covering space for all the genus g ≥ 2 surfaces. Correspondingly, all genus g ≥ 2 surfaces can be obtained from the upper half-plane by dividing by the action of some discrete group. The elements of this group must be isometries of the metric (9.2.3); these isometries are just the M¨ obius transformations with real coeﬃcients, P S L(2, R). Figure 9.1 shows the genus-three surface with some closed curves drawn in, an A-cycle and B-cycle for each handle. These are nontrivial curves, in that they cannot be contracted to points. If we traverse one of these curves, on the covering space we come back not to the original point but to one of those identiﬁed with it. Thus to each of the cycles is associated an element of P S L(2, R), also denoted Ai or Bi . The A- and B-cycles are complete, in that every closed path is equivalent (homotopic) to a succession of A- and B-cycles. In fact they are slightly overcomplete: the path −1 −1 −1 −1 (A1 B1−1 A−1 1 B1 )(A2 B2 A2 B2 ) . . . (Ag Bg Ag Bg )

(9.2.4)

can be contracted to a point, and the corresponding product must be the identity. Counting moduli, there are three real parameters in each P S L(2, R) element, giving 6g, minus three from the constraint (9.2.4),

9.2 Higher genus Riemann surfaces

293

minus three more because overall conjugation by P S L(2, R) gives an equivalent surface, for 6g − 6 in all. Although the Schottky and Fuchsian parameterizations are similar, they have rather diﬀerent properties. In particular, the complex structure is much less evident in the latter, and the extension to the superstring correspondingly less simple. The period matrix The modulus τ of the torus has a natural g × g matrix generalization at genus g. To see this, we need ﬁrst the Abelian diﬀerentials or holomorphic one-forms. These are world-sheet ﬁelds ωa with the property that in complex coordinates ∂¯z ωz = 0 ,

ω¯z = 0 .

(9.2.5)

There are precisely g such ﬁelds, and g more satisfying the conjugate relation. This follows from the Riemann–Roch theorem (5.3.22). The Abelian diﬀerentials (9.2.5) and their conjugates are just the kernel of the operator P0T , for which the Riemann–Roch theorem is dim ker P0T − dim ker P0 = −χ = 2g − 2 .

(9.2.6)

The operator P0 acts on scalar ﬁelds c and ˜c with h = 0, and so reduces to the ordinary gradient. This always has two zero modes (c = constant or ˜c = constant), and so dim ker P0T = 2g. The line integral of a one-form is a coordinate-invariant quantity. A natural basis for the Abelian diﬀerentials is then ωzi , i = 1, . . . , g, deﬁned by

Ai

dz ωzj = δij .

Then the period matrix

(9.2.7)

τij =

Bi

dz ωzj

(9.2.8)

characterizes the complex structure of the surface. At genus one, where ωz is constant, this is the usual τ. It can be shown that τij is symmetric, so it has 12 g(g + 1) complex parameters. For g = 1, 2, 3 this is equal to the number of complex moduli, but for g ≥ 4 it is greater, so not all period matrices correspond to Riemann surfaces. The theta function (7.2.36) has a natural genus-g generalization, where τ becomes the period matrix and ν, n, a, b become g-component vectors.

294

9 Higher order amplitudes Hyperelliptic surfaces

Here is another description which is sometimes useful. Consider the space S2 × S2 , with coordinates z and y. Consider the two-dimensional submanifold given by the equation y2 =

2k

(z − Zi )

(9.2.9)

i=1

for complex parameters Zi . Projecting down to the z-coordinate, there are two sheets joined across k branch cuts. This produces a surface with g = k − 1 handles. The branch points are singular when projected down to z, but the full surface (9.2.9) is nonsingular there: y is locally a good coordinate. The surface depends on the 2k complex Zi , but three of these are again redundant under a M¨ obius transformation, leaving 2k−3 = 2g−1 complex parameters. For g = 1, 2 this is the same as the number of moduli, but for g ≥ 3 the number of moduli is greater and only special surfaces have the form (9.2.9). These are known as hyperelliptic surfaces. 9.3

Sewing and cutting world-sheets

A useful concept for analyzing string amplitudes is the relation between higher and lower genus amplitudes. A key ingredient is the plumbing ﬁxture, for building up higher genus surfaces from lower ones. Consider two world-sheets, M1 with coordinate z1 in some patch, and M2 with coordinate z2 in some patch. We can join these to form a new worldsheet M as follows. Let q be a complex parameter. Cut out from M1 the disk |z1 | < (1 − )|q|1/2 for some small , and from M2 the disk |z2 | < (1 − )|q|1/2 . Now form a single surface by identifying pairs of points such that z1 z2 = q .

(9.3.1)

This is shown in ﬁgure 9.2. The annular region (1 − )−1 |q|1/2 > |z1 | > (1 − )|q|1/2

(9.3.2)

is sewn as indicated. One notation for this construction is M = M1 ∞M2 (z1 , z2 , q) .

(9.3.3)

The notation emphasizes that the resulting surface depends on the choice of local coordinates. With diﬀerent local coordinates z1 and z2 , one would be cutting out diﬀerent circles and the joined surface will in general be diﬀerent. The two patches might also be in diﬀerent parts of a single

295

9.3 Sewing and cutting world-sheets

M1

M2

M Fig. 9.2. Joining two world-sheets by removing a disk (shaded) from each and gluing the edges as indicated by the dashed lines.

surface M0 . The plumbing ﬁxture construction then adds a handle to the surface. This is denoted M = M0 8(z1 , z2 , q) .

(9.3.4)

The symbols ‘∞’ and ‘8’ are supposed to depict the two sewing processes. The parameter q is not actually needed — taking z1 = z1 q −1/2 , z2 = z2 q −1/2 , and q = 1 gives an equivalent surface. It is useful to introduce q in order to discuss the behavior as a function of q with ﬁxed coordinate patches. We can also reverse this construction. Given any Riemann surface M, and any non-self-intersecting curve C on it, one can always ﬁnd holomorphic coordinates z1 = z2−1 in a neighborhood of C such that C is given by |z1 | = |z2 | = |q|1/2 . Cutting on C (undoing the identiﬁcation z1 z2 = q) and gluing in the disks |z1 | < |q|1/2 and |z2 | < |q|1/2 produces M1 and M2 (or M0 ) such that M would be reconstructed by the sewing operation — either (9.3.3) or (9.3.4) depending on whether the cut along C makes the surface disconnected. For the genus-two example in ﬁgure 9.2, consider the case that q 1. Figure 9.3(a) shows M in this case. We have smoothly interpolated the metric between dz1 d¯z1 and dz2 d¯z2 in the overlap region. The surface

296

9 Higher order amplitudes

(a)

(c)

(b) Fig. 9.3. Three representations of the genus two surface with degenerating cycle.

pinches down to a radius of order |q|1/2 . By a Weyl transformation, one can ﬁnd a metric in which the radius stays of order 1, as shown in ﬁgure 9.3(b). Since a Weyl transformation acts isotropically, the eﬀect is that the length of the handle becomes large, of order − ln |q|. The handle region |z1 | < 1, |z2 | < 1 is equivalent to the cylinder ln |q| < Im w < 0 ,

w∼ = w + 2π ,

(9.3.5)

where z1 = exp(−iw) and z2 = q exp(iw). Figure 9.3(c) shows one more representation of the same surface. The annular region O(|q|) < |z| < O(1) surrounding the small handle is conformally equivalent to the long cylinder of ﬁgure 9.3(b). In terms of ﬁgure 9.3(a), one can think of this picture as a Weyl transformation where all of M2 has been reduced. In all the pictures, it is clear that the surface becomes special in the q → 0 limit. The handle is said to pinch or degenerate. This is an example of a boundary of moduli space. The boundary of moduli space is important, because any divergences must come from there. In fact, we will see that every boundary is of this form, where one or more handles pinches. For the genus-two surface, there is one other way for a single handle to degenerate, if one of the A- or B-cycles pinches. This would correspond to building the genus-two surface from a single torus via the ‘8’ operation (rather than from two tori via the ‘∞’ operation) and taking q → 0. It is a useful fact that by repeated use of the cutting operation, every Riemann surface can be reduced to spheres with three punctures. A puncture is a marked point, with a speciﬁed complex coordinate near

9.3 Sewing and cutting world-sheets

(a)

297

(b)

(c) Fig. 9.4. Some sewing constructions of the genus-two surface with four operators. Each vertex is a sphere with three punctures, each internal line represents the plumbing ﬁxture construction, and each external line represents a vertex operator.

the point; this is the data needed to sew in a plumbing ﬁxture. In other words, every Riemann surface can be constructed from spheres with three punctures via sewing. For example, ﬁgure 9.4 shows three of the many ways to construct the genus-two surface with four vertex operators. There is one complex modulus q for each sewing operation, giving seven in all for each of the constructions in ﬁgure 9.4. These correspond to the 3g − 3 = 3 complex moduli for the genus-two surface plus the positions of the four operators. A graphical decomposition The graphs in ﬁgure 9.4 are reminiscent of Feynman diagrams. We now describe a Feynman-diagram-like decomposition of moduli space. The vertices are rather more complicated than in ﬁgure 9.4, with any number of external lines and with internal corrections from string loops. The point of the construction is to make explicit the nature of the boundary of moduli space. The statement is that the integral over moduli space is given by a sum of Feynman graphs. Each internal propagator in a graph represents the sewing operation, with the sewing parameter q integrated over some standard region such as |q| < 1. The n-point vertex represents a Riemann surface with n punctures. We will see that the surfaces in the vertices must be integrated over some interior region of moduli space, and also summed over genus. The sewing operation refers to a speciﬁc complex

298

9 Higher order amplitudes

Fig. 9.5. Trilinear string vertex, with three marked points and three sewing circles on the sphere.

(a) 1 z4

(c)

(b)

Fig. 9.6. (a) Constructing the sphere with four vertex operators. (b) The region of the z-plane covered by this graph. (c) The additional vertex needed to give the correct amplitude. The four-punctured sphere is integrated over the unshaded region.

coordinate for each point to be sewn, so the deﬁnition of the vertex must include such a choice at each point. The set of genus-g Riemann surfaces appearing in the n-point vertex is denoted Vg,n . External lines represent vertex operators. Let us illustrate this construction. Start with a three-point vertex V0,3 , shown in ﬁgure 9.5 as a sphere with three marked points and three sewing circles. Sewing circles are unit circles in the complex coordinates associated with the respective points. Build a sphere with four points by taking two of these vertices, removing the interior of one sewing circle from each, and sewing them to the ends of a cylinder; this is equivalent to the plumbing ﬁxture construction. This corresponds to the Feynman graph of ﬁgure 9.6(a); the length of the cylinder (with the circumference scaled to

9.3 Sewing and cutting world-sheets

299

2π) is − ln |q|. Integrating over |q| < 1 and summing over the other two channels gives only part of the integration region for the sphere with four points, as shown in ﬁgure 9.6(b). The shaded regions are included but the unshaded interior region is missed. Let us make this more explicit. For the trilinear vertex, put the points on the sphere at the standard 0, 1, ∞ and choose coordinate frames z (0) , z (1) , and z (∞) at each, so the three sewing circles are |z (0,1,∞) | = 1. The choice should be invariant (up to phase) under the six M¨ obius transformations that permute the vertex operators, so that the vertex is symmetric. There are many possible choices, for example az a(z − 1) a , z (1) = , z (∞) = , (9.3.6) z (0) = z−2 z+1 2z − 1 for any constant a. For a ≥ 3, the sewing circles do not overlap, guaranteeing that the plumbing ﬁxture produces a good surface for all q ≤ 1. As the reader can verify, sewing the z (∞) and z (0) patches from two spheres with parameter q produces a sphere with vertex operators at 0, 1, 12 (1 + a2 /q), and 12 (1 − a2 /q). This is equivalent under a M¨ obius transformation to vertex operators at 0, 1, ∞, and z4 = (q − a2 )2 /4a2 q .

(9.3.7)

As q runs over the unit disk, z4 covers the outer shaded region of ﬁgure 9.6(b). Permuting the external vertex operators gives the other two regions. We could try to take a smaller, so the shaded regions grow, but they will eventually overlap and double-count some surfaces while other surfaces will still not be produced. They will never precisely ﬁll the plane, so we need to add in an explicit four-point vertex V0,4 which is given by the sphere with four marked points, integrated over the unshaded part of the z-plane. This does not deﬁne V0,4 completely because there is also a choice of coordinate at each puncture for each value of z4 . It is always possible to make this choice in such a way as to connect smoothly at the boundary between the shaded and unshaded regions. Note that the Feynman graph of ﬁgure 9.6(a) includes all the asymptotic regions z4 → 0, 1, ∞, while the four-point vertex includes only nondegenerate surfaces. The behavior in the limits is then given explicitly by the q → 0 limit of the plumbing ﬁxture: this is why this construction is useful. As another example, sewing two of the circles of a single vertex to the ends of a cylinder produces a torus with one vertex operator. This covers the limit Im τ → ∞ of the moduli space of the torus, but an interior region is missed and must be added by hand as a vertex region V1,1 . This construction can be extended inductively. The sewing operation always increases the parameter r = 3g + n − 2 .

(9.3.8)

300

9 Higher order amplitudes

The vertex V0,3 , the original sphere with three punctures, has r = 1. The 8 operation gives r = r0 + 1, while the ∞ operation gives r = r1 + r2 . Thus, V0,4 and V1,1 both have r = 2. It has been shown that one can construct Vg,n inductively in r, and so cover moduli space as claimed. A more explicit construction is as follows (the proofs can be found in the references). On any Riemann surface there is a unique minimal area metric, deﬁned as the metric of least area subject to the constraint that every nontrivial closed curve (those which cannot be continuously contracted to a point) has length at least 2π. Every point on the surface lies on at least one saturating geodesic, a nontrivial closed curve of length exactly 2π. Saturating geodesics that can be continuously deformed into one another (that is, are homotopically equivalent) cannot intersect. Each set of homotopically equivalent saturating geodesics forms a band (topologically an annulus) on the surface. The diﬀerent bands cover the surface completely, and may overlap. The height h of such a band is deﬁned as the minimum distance between its two bounding geodesics. The idea of a minimal area metric can be extended to surfaces with punctures, with the speciﬁcation that a curve that circles a puncture is regarded as nontrivial. The minimal area metric near a puncture is of the form dzd¯z /z¯z , a semiinﬁnite cylinder of radius 2π. There are thus two kinds of bands: the ﬁnite ones in the interior of the surface, and the semi-inﬁnite ones centered on the punctures, terminating in the interior of the surface. We can now describe Vg,n in this construction. Take all Riemann surfaces with n punctures, with the restriction that the height of any internal band, in the minimal area metric, is less than 2π. Cut oﬀ the semi-inﬁnite cylinders, leaving a stub of length π, and glue in the usual unit disk with vertex operator in the end of the stub. With these vertices, the Feynman graph sum covers moduli space exactly once. It can be shown that the sewing of these minimal area surfaces still produces a minimal area surface. Bands of height less than 2π are explicitly included in the vertices. Bands of height greater than 2π are produced by the sewing operation on the stubs, the resulting height being 2π from the stubs plus − ln |q| from the plumbing ﬁxture; this runs from 2π to ∞ when q is integrated. The minimum length and maximum height restrictions guarantee that degenerating surfaces do not appear in the vertices, so they arise only from one or more qs going to zero, as claimed. 9.4

Sewing and cutting CFTs

We have described higher genus Riemann surfaces as built up by sewing together lower genus surfaces. Using the sewing properties of path integrals, the CFT on the higher genus surface can similarly be related to

301

9.4 Sewing and cutting CFTs

Σi

Ai A

(a)

i

M1

Σi

M2

AiA

(b)

i

M1

M2

Fig. 9.7. (a) The path integral on M is cut open by inserting a complete set of boundary states. (b) Each state is replaced by a disk with local operator.

lower order amplitudes. The relation is shown pictorially in ﬁgure 9.7. For clarity we ﬁrst discuss the case q = 1. The path integral on M is cut open on the circle |z1 | = |z2 | = 1, inserting a complete set of boundary states. Each boundary state is then replaced by a disk with the corresponding local operator, leaving the uncut world-sheets M1 and M2 . The precise form of the result is

. . .1 . . . 2 M =

ij

. . .1 Ai(z1 )

M1

Gij Aj(z2 ) . . .2

M2

.

(9.4.1)

We are allowing arbitrary additional insertions . . .1 on M1 and . . .2 on M2 ; these must of course lie outside the circles |z1 | = 1 and |z2 | = 1 that are being cut. We have not speciﬁed any particular orthonormality for the basis; the appropriate inverse metric Gij is to be determined. The insertions Ai and Aj that replace the handle are implicitly at the origin in the z1 and z2 frames respectively. Now, let us consider the case in which M2 is a sphere, with the sewing coordinate z2 being u and with . . .2 being some local operator Ak at z = 0. We have the identiﬁcations z1 u = 1 from the plumbing ﬁxture and zu = 1 from the sphere, so we can just extend z1 = z through the whole sewn region. The sewn surface M is the same as M1 with an additional

302

9 Higher order amplitudes

operator Ak at z1 = 0. Then eq. (9.4.1) becomes

. . .1 Ak(z1 )

M1

=

1) . . .1 A(z i

M1

ij

Gij Aj(u) Ak(z)

S2

(9.4.2)

.

The two-point function Aj(u) A(z) k S2 is just the inner product

j|k = Gjk . Eq. (9.4.2) then implies Gij Gjk = δ ik

(9.4.3)

Gij = Gij .

(9.4.4)

or

That is, the inner product deﬁned by the two-point function on the sphere is the correct one for sewing. We have thus derived the relation (9.4.1) for the case in which M2 is a sphere with an arbitrary local operator. By the state–operator isomorphism, any boundary state can be obtained in this way. Since the path integral is local, the result depends only on the region of the sewing and not on the rest of the surface, so it holds for arbitrary M2 and . . .2 . Sewing with parameter q is the same as sewing with q = 1 and local coordinates z1 and z2 = z2 /q. We thus obtain eq. (9.4.1) with Aj in the primed frame, and so obtain an extra factor from the conformal transformation back to z2 , (z )

˜

Aj 2 = q hj q¯hj Aj(z2 ) . The result is

. . .1 . . . 2 M =

˜

q hj q¯hj

. . .1 Ai(z1 )

ij

(9.4.5)

M1

Gij Aj(z2 ) . . .2

M2

,

(9.4.6)

where we have taken a basis diagonal in the weights. For the ‘8’ operation the path integrals on M and M0 are related in the same way:

. . . M =

ij

˜

(z2 ) 1) q hj q¯hj Gij . . . A(z i Aj

M0

.

(9.4.7)

It is interesting to consider again the boundary of moduli space, given by the q → 0 limit of one or more handles. We see that the operators of lowest weight will control the behavior. The results (9.4.6) and (9.4.7) resemble the OPE, and in fact (9.4.6) reduces to the OPE when M2 is a sphere and ‘. . .2 ’ is a pair of vertex operators. This analogy is especially clear from ﬁgure 9.3(c), representing the limit as the insertion of a small handle. The sewing relation shows that not only a product of operators, but also a small handle, or any other disturbance that we might sew into a small circle on the world-sheet, is equivalent to a sum of local operators. The result (9.4.6) is an elegant generalization of the OPE. From the point

303

9.4 Sewing and cutting CFTs

of view of M1 , there is a tiny copy of M2 glued in near z1 = 0. This can be expanded in local operators Ai , with coeﬃcient functions Ai . . .2 M2 . From the point of view of M2 , there is a tiny copy of M1 glued in near z2 = 0, which is expanded in local operators Ai , with coeﬃcient functions

. . .1 A i M 1 . We have argued that every Riemann surface can be reduced to thricepunctured spheres. The sewing construction thus relates all amplitudes to the three-point amplitudes on the sphere, which are essentially the operator product coeﬃcients: these are the data that deﬁne the CFT. We will encounter many equivalences between CFTs — we have already seen T -duality, and bosonization (chapter 10) will be another notable example. The sewing construction provides the criteria for two theories to be the same: a one-to-one correspondence between their states (or local operators) and identical OPEs.1 In fact, as we will show in chapter 15, the operator products of the primary ﬁelds alone determine those of all the others. Let us explore this point of view further. One can cut a surface along many diﬀerent sets of curves, so there must be consistency conditions to guarantee that the sewing relation always gives the same result for the expectation values on M. Consider ﬁrst an inﬁnitesimal change in the cutting curve C. A general inﬁnitesimal change leaving M invariant can be obtained by an inﬁnitesimal change in the coordinates used for the sewing, z1

= z1 +

z2 = z2 −

∞ n=−∞ ∞

n z1n+1 ,

(9.4.8a)

−n q −n z2n+1 .

(9.4.8b)

n=−∞

The diﬀerence of the sewn expressions is then proportional to (z )

(z )

Ai 1 Gij Aj 2 − Ai(z1 ) Gij Aj(z2 ) = −

∞

(z2 ) 1 ) ij n Ln ·A(z i G Aj

n=−∞ 2) −Ai(z1 ) Gij L−n ·A(z j

+ (anti) , (9.4.9)

where we have used the conformal transformation (2.9.6), and ‘anti’ de˜ n . The vanishing of the notes the same form with n → ∗n and Ln → L quantity in braces can be shown by an argument parallel to that used to derive eq. (9.4.4). This is just the statement that one can pull the contour 1

We should note that when there is more than one (2, 0) plus (0, 2) ﬁeld that could play the role of Tab , as in the free-ﬁeld examples of section 2.5, it is necessary to specify which is the energy-momentum tensor. This choice determines the conformal transformations of operators, which enter implicitly into the sewing relation.

304

9 Higher order amplitudes

integral that deﬁnes the generator through the sewn handle; the change from Ln to L−n is from an inversion of coordinates. All other contour integrals for conserved charges can be pulled through as well. There are also sets of curves that are not related by a continuous deformation. The four-point amplitude on the sphere can be written in terms of three-point functions by cutting on curves circling diﬀerent pairs of operators. Equality between diﬀerent sewings is the associativity of the OPE, ﬁgure 2.8. Another example is given by the torus: we can reduce the torus to the sphere by cutting on the A-cycle, or the B-cycle, or any other nonself-intersecting closed curve. In terms of the plane with identiﬁcations, a closed curve C that begins at (σ 1 , σ 2 ) = (0, 0) must end at an identiﬁed point 2π(m, n) .

(9.4.10)

All curves with the same m and n values can be continuously deformed into one another, so we need to consider the relation between sewing on curves of diﬀerent m and n. One can show that for a nontrivial non-selfintersecting curve, m and n must be relatively prime, which implies that there are integers p and q such that mp − nq = 1. There is then a modular transformation (σ 1 , σ 2 ) = (σ 1 , σ 2 )

m n q p

,

(9.4.11)

such that the curve C comes back to (σ 1 , σ 2 ) = 2π(1, 0) ,

(9.4.12)

which is the A-cycle. So cutting on a general nontrivial non-self-intersecting curve is equivalent to cutting on the A-cycle for a modular-equivalent value of τ. The consistency of the sewing construction is thus equivalent to modular invariance. Every set of sewing curves may be transmuted into any other by a series of two moves: the four-point associativity transformation on the sphere and the modular transformation on the torus with a single vertex operator. Figure 9.4 makes this plausible. For example, ﬁgures 9.4(a) and 9.4(b) are related by one associativity condition. Repeated applications of associativity can bring the sewing to the form of ﬁgure 9.4(c), where all loops are expressed in terms of the one-loop, one-point function. So these are the conditions for an arbitrary set of operator product coeﬃcients to deﬁne a sensible conformal ﬁeld theory. The program of solving the duality constraints is known as the conformal bootstrap. It has been carried out only in cases where additional restrictions have been introduced, as we will discuss in chapter 15.

305

9.5 General amplitudes

All the considerations of this section extend to world-sheets with boundary, where in addition to degenerating handles there may be degenerating strips. The consistency conditions have been generalized to this case. 9.5

General amplitudes

We now have assembled all the ingredients that we need in order to understand the consistency of general string amplitudes. Having given a graphical decomposition of moduli space, the sewing formalism allows us to give a diagrammatic expression for the general string amplitude. Consider ﬁrst the propagator, from the plumbing ﬁxture. The ghost insertion for the modulus q is

dz ∂z1 b0 = bz1 z1 , 2πi ∂q z2 q

(9.5.1)

q . Including these insertions, and using the and similarly for q¯ it is ˜b0 /¯ sewing expression (9.4.6), the factors associated with the propagator are 8πδm2 ,m˜ 2 Gij b0 ˜b0 d2 q α (k2 +m2 )/4 α (k2 +m˜ 2 )/4 ij ˜ i i i i . (9.5.2) G b0 b0 = 2 i i 2 q¯ q q α (ki + mi − i) |q|<1 q¯ The n-string coupling is given by inserting vertex operators into the surfaces appearing in the vertices of the sewing construction, g(i1 , . . . , in ) =

g

m

d t

Vg,n

m

k=1

Bk

n l=1

ˆi V l

.

(9.5.3)

g

where m = 6g + 2n − 6 is the number of moduli. The Feynman graphs constructed from the couplings (9.5.3) and propagators (9.5.2) then reproduce the full string amplitude. The label i again stands for the state and momentum of each string. The tree amplitude in section 9.1 illustrates this structure. The ﬁrst issue is the ﬁniteness of the amplitude. We assert that divergences can arise only from the boundary of moduli space, as in the various low order examples. At ﬁnite values of the moduli, the sewing formula gives a sum over an inﬁnite number of string states, but this sum converges because it is ultimately just the expansion of a quantum state in a complete set. Actually, convergence is guaranteed only in a unitary CFT, but the nonunitary parts of the string CFT are quite tame. The X 0 CFT is deﬁned by analytic continuation from a unitary CFT, while the ghost CFT just generates a Faddeev–Popov determinant and so is ﬁnite if we have made a good choice of gauge. Now, the boundary has been made explicit as the q → 0 limit of the various propagators, and the integral (9.5.2) is one that we have encountered several times before. It is

306

9 Higher order amplitudes

deﬁned via analytic continuation, giving an ordinary ﬁeld-theoretic propagator. The only singularities are then the poles in propagators, which represent production of on-shell particles in the intermediate states. In particular they are long-, not short-, distance divergences. The other issue is unitarity. We studied this for the four-point tree amplitude in some detail. Having brought the general amplitude to a Feynman-diagram form, rather than work out the combinatorics in detail we will simply cite standard ﬁeld theory results on the unitarity of the diagrammatic expansion. Also, the demonstration at the end of section 9.1 that the sum over intermediate states reduces to a sum over physical states generalizes as in ﬁeld theory. In summary, the issues involved in the perturbative ﬁniteness and unitarity of string theory appear to be well understood. Observe once again that we have not assumed 26 ﬂat dimensions, but merely d ﬂat dimensions plus a compact positive norm CFT of central charge 26 − d. Of course, any discussion of the unitarity of the ﬂat spacetime S-matrix for the bosonic string is somewhat formal, since ﬂat spacetime is not a solution of the theory — ﬁrst there is the tachyon instability, then the cosmological constant induced by loops. An alternative approach to the ﬁniteness and unitarity of string theory, based on the light-cone gauge, has also been extensively developed. This gauge gives up covariance, but has certain advantages — it makes possible a rather explicit description of moduli space, and the amplitudes can be represented in a ﬁeld-theoretic form with a Hamiltonian that is local in the light-cone time x+ . Covariance has then been established by showing that the perturbation theory is equivalent to the covariant perturbation theory. String divergences Before moving on, let us expand somewhat further this point that string divergences are long-distance eﬀects. Consider the one-loop amplitude with n vertex operators, taking the limit that m operators come together with τ held ﬁxed as in ﬁgure 9.8(a). The string loop of course includes gravity, and ﬁgure 9.8(a) is reminiscent of the graviton loop of ﬁgure 1.1(c), which was UV divergent when the interaction vertices came together in spacetime. There is indeed a divergence in this string amplitude, but it is from long spacetime distances, not short ones. Figure 9.8(b) shows a conformally equivalent picture where the m vertex operators are connected to the rest of the torus by a long cylinder. Cutting the cylinder open and inserting a complete set of states, we obtain the usual expression (9.5.2), with q → 0 being the limit where the vertex operators become coincident. So the only singularities are the usual poles from propagation over long spacetime

9.5 General amplitudes

(a)

307

(c)

(b) Fig. 9.8. (a) Torus with n vertex operators, in the limit that m of the operators come together. (b) Conformally equivalent picture: the m vertex operators at the end of a long cylinder. (c) Sewing construction, in terms of the product of a sphere with m + 1 vertex operators and a torus with n − m + 1.

distances. Conformal invariance once again changes short distances on the world-sheet into long distances in spacetime. The cases m = n and m = n − 1 present special problems. When m = n, meaning that all the vertex operators are coming together, momentum conservation forces k µ to vanish. For massless states, the pole is then 1/0 and the usual analytic continuation does not help because momentum conservation forces the amplitude to sit right on the massless pole. This is a tadpole, just as in section 7.4, and the resolution is the same, a correction to the background ﬁelds. The divergence here is proportional to the torus amplitude with one vertex operator. In the other problematic case, m = n − 1, all the vertex operators but one, say Vn , come together. Momentum conservation requires that the momentum ﬂowing through the cylinder be k µ = −knµ . However, conformal invariance requires that kn2 = −m2n , so the intermediate momentum is ﬁxed on top of a pole for those intermediate states with m2j = m2n . Once again, there is an analogous divergence in ﬁeld theory, shown in ﬁgure 9.9: the internal propagator is 1/0. This divergence occurs whenever the mass of the internal line j is the same as that of the on-shell external particle n. The interpretation here is simple. The loop represents a radiative correction to the mass of the particle n. The S-matrix is deﬁned with the external momentum taken on the true mass shell, including radiative corrections, and it is again free of this IR divergence. Note that the issue here is not some UV divergence internal to the loop, but the IR divergence on the

308

9 Higher order amplitudes

* Fig. 9.9. Field-theory graph analogous to the case m = n − 1. The kinematics force the starred propagator to diverge. The interpretation is a (ﬁnite) mass shift due to the loop.

starred propagator which appears even when the mass correction is ﬁnite. The resolution in the string case is the same. The momentum appearing in the vertex operator must be at the true physical mass; this is not in general the same as the tree-level mass, the one whose square was always a multiple of 1/α . Note that for certain massless ﬁelds — in particular gauge bosons — gauge invariance guarantees that they will remain massless to all orders. Incidentally, for massive states the mass shift is usually complex, because these states are unstable to decay into lighter string states. There is a puzzle here. We saw early on that consistency of string theory, the condition that the world-sheet theory be conformally invariant, required the beta functions (3.7.14) to vanish. These beta functions contain no information about higher genus surfaces — they correspond to the treelevel string ﬁeld equations. What happens when we try to expand around a loop-corrected background? In fact things work out just right. The dangerous limit in ﬁgure 9.8 gives rise not only to a divergence, but also to a conformal anomaly. To see this heuristically, picture the limit as in ﬁgure 9.3(c), as a small handle glued into the sphere. The small handle can be represented by a series of local operators, and in particular from the massless level there is q ¯ µ dqd¯ . (9.5.4) ∂X µ ∂X q¯ q Now, cut oﬀ the divergence in a coordinate-invariant way by requiring the size of the handle to be greater than a; that is, |q|eω > a, where the metric on the sphere is dzd¯z e2ω . The lower limit on the integral (9.5.4) gives ¯ µ ln(ae−ω ) . (9.5.5) − 4π∂X µ ∂X Thus we have new dependence on the Weyl factor of the metric: this G , eq. (3.7.14). To obtain a Weylcorrects the sigma-model Weyl anomaly βµν invariant theory, the background ﬁelds must satisfy the full string-loopcorrected Weyl invariance conditions (or, equivalently, BRST invariance). The Weyl anomaly from the shifted background cancels that from the small handle. This is the Fischler–Susskind mechanism. From ﬁgure 9.3(c),

9.5 General amplitudes

309

we can think of the divergence and the Weyl anomaly as world-sheet UV eﬀects, not from the usual short-wavelength ﬂuctuations of the ﬁelds, but from a small topological ﬂuctuation. We should emphasize that on a ﬁxed Riemann surface of any genus, the local structure is unaﬀected by the topology; it is the integration over Riemann surfaces that produces the divergence and Weyl anomaly. This can also be regarded as a breakdown of the canceled propagator argument. Because the kinematics forces the amplitude to be at the pole, we cannot appeal to analytic continuation to eliminate possible surface terms under a BRST variation. In fact they appear, corresponding to the Weyl anomaly (9.5.5). In the loop-corrected background, the BRST anomaly from the shift of the background cancels that from the boundary of moduli space — the stringy technicalities all ﬁt together to give a consistent spacetime picture. To implement this in string perturbation theory is rather clumsy, but fortunately it is rarely necessary. First, in most supersymmetric theories the backgrounds are uncorrected. Second, for the analysis of long-distance physics in string theory it is usually most eﬃcient to proceed in two steps, ﬁrst using string theory to derive the form of the low energy eﬀective action and then using the eﬀective ﬁeld theory at long distance. One then does not need to deal with the IR divergences at the string level. In the present case, the tadpoles imply a potential energy term δS = −

Λχ

dd x (−G)1/2 e−χΦ

(9.5.6)

χ

in the eﬀective action. Here Λχ is the tadpole from Euler number χ, and the dependence of the action on Φ follows from the fact that eΦ is the string coupling. The separate tadpoles for the graviton and dilaton can be extracted from the string amplitude, with some eﬀort. The same idea extends to the vertex operator correction from the mass shift, and to other IR divergences such as those of exclusive amplitudes in theories with massless particles. These ﬁeld-theory divergences must also appear in the corresponding string-theory amplitudes, since string theory reduces to ﬁeld theory at long distance. Again we emphasize that while UV divergences in ﬁeld theory generally signify a fundamental limitation on the range of validity of the theory, IR divergences usually mean that one is simply asking the wrong question (wrong background, wrong mass, exclusive versus inclusive cross-section, etc.) in what is otherwise a perfectly consistent theory. Any such divergences in string theory should then be dealt with precisely as in ﬁeld theory. Let us review what we have learned. Modulo problems particular to the bosonic string, the S-matrix is ﬁnite and unitary. The basic mechanism is modular invariance, which cuts oﬀ the UV divergences without spoiling

310

9 Higher order amplitudes

the spacetime gauge invariance. Of course, knowing that string theory makes sense in weak coupling perturbation theory around ﬂat spacetime is hardly enough. We would like to know that the theory makes sense beyond perturbation theory, and have some means of calculating nonperturbative corrections. Also we would like to know what string theory implies for the nature of spacetime at short distance. In the remainder of this chapter we brieﬂy describe some of the attempts that have been made in these directions. 9.6

String ﬁeld theory

It is interesting to compare the approach we have taken in string theory to the way quantum ﬁeld theory is formulated. The world-sheet path integral is analogous to a sum over particle paths, often called the ﬁrstquantized description. In quantum ﬁeld theory, on the other hand, we start with a path integral (or its Hamiltonian equivalent) over spacetime ﬁelds. One could instead develop quantum ﬁeld perturbation theory in the ﬁrst-quantized form; it is necessary to introduce explicit interaction vertices, spacetime points where n particle paths end. In string theory this second step is not necessary — there are no special points, and the sum over topologies produces the interactions. However, the result is only the perturbation expansion, and there are important phenomena that are not contained in this expansion. In the Standard Model, these include quark conﬁnement and the ﬁnite-temperature baryon number violation of the weak interaction. There does not seem to be any useful sense in which these are contained in a resummation of the perturbation series; rather they require a more general formulation of the theory. In the previous section we have expressed string amplitudes in a propagator and vertex form that looks like the perturbation theory for an inﬁnite-component ﬁeld theory. Could we instead derive this from a path integral over an inﬁnite set of spacetime ﬁelds? And is there some principle, extending gauge invariance and general coordinate invariance, that determines the form of the theory? This is the subject of string ﬁeld theory. We will be able to sketch here only a few of the ideas, ones that are particularly close to the approach that we have been taking. The reader should consult the literature for more details. We will start with the guess that we should take the string wavefunctional Ψ[X] and elevate it to a ﬁeld operator, or equivalently to a path integral variable of integration — so-called second quantization. Now, Ψ[X] depends on the curve X µ (σ) traced out by the string in spacetime, and roughly speaking creates or destroys a string in this conﬁguration. However, we need to be more precise. For example, should we require Ψ

9.6 String ﬁeld theory

311

to be invariant under reparameterization of the curve, and should it also depend on the Polyakov metric gab ? Things seem to work much more nicely if one starts with the BRST-invariant form of the theory, including the Faddeev–Popov ghosts, and then second quantizes. From the commutation relations, b and c are conjugate to one another (as are ˜b and ˜c), so if for example we treat c and ˜c as coordinates the wavefunctional is Ψ[X, c, ˜c ]. We then have a path integral running over all such functionals,

[dΨ] exp(iS [Ψ]) .

(9.6.1)

We will consider ﬁrst the open string, describing one particular approach. What symmetry principle will restrict the string action? From our study of the string vertex operators and null states, we might guess the symmetry δΨ = QB Λ

(9.6.2)

for any functional Λ. There is an extremely simple action with this invariance, S0 =

1

Ψ|QB |Ψ . 2

(9.6.3)

The equation of motion is just QB Ψ = 0 .

(9.6.4)

One way to see that this is reasonable is by expanding the string ﬁeld Ψ in terms of an inﬁnite number of ordinary ﬁelds, one for each internal state of the string. Let Ψi [X , c, ˜c ] be a complete set of wavefunctions for the internal modes — all but the center-of-mass xµ . Then a general state can be expanded: Ψ[X, c, ˜c ] =

Φi (x)Ψi [X , c, ˜c ] ,

(9.6.5)

i

the expansion coeﬃcients being functions of the remaining variables xµ . The string ﬁeld path integral becomes an inﬁnite product of path integrals over the component functions,

[dΨ] →

[dΦi ] .

(9.6.6)

i

As usual, we can start with the ground state functional Ψ0 and build the remaining states with the raising operators. For example, keeping all

312

9 Higher order amplitudes

(a)

(b) 1

2

Fig. 9.10. (a) Ψ1 ∗ Ψ2 , convoluting the right half of Ψ1 with the left half of Ψ2 . One can think of this as deﬁned by a path integral over the shaded region, whose width is then taken to zero. (b) Ψ, convoluting the right and left halves of Ψ.

states at the ﬁrst two levels,

Ψ = ϕ(x) + Aµ (x)αµ−1 + B(x)b−1 + C(x)c−1 + ϕ (x)c0

+ Aµ (x)αµ−1 c0 + B (x)b−1 c0 + C (x)c−1 c0 + . . . Ψ0 . (9.6.7) Here, ϕ is the tachyon ﬁeld and Aµ the gauge ﬁeld, while B and C are the Faddeev–Popov ghost ﬁelds associated with the spacetime gauge symmetry. The primed ﬁelds can all be gauged away or are auxiliary (that is, they satisfy algebraic rather than diﬀerential equations and can be eliminated). One can then check that the symmetry (9.6.2) includes the ordinary gauge transformation on Aµ , while the action is of Klein–Gordon form for ϕ and of Maxwell form for Aµ . The BRST string ﬁeld formalism thus gives a very compact generalization of spacetime gauge invariance. Interactions also work out rather nicely. Deﬁne a product operation Ψ1 ∗Ψ2 on string ﬁelds by the functional integral depicted in ﬁgure 9.10(a), forming the overlapof the right half of Ψ1 with the left half of Ψ2 . Deﬁne also an operation , giving the overlap of the right and left halves of the single string wavefunctional as shown in ﬁgure 9.10(b). To make an interacting theory, add a nonlinear term to the gauge transformation, δΨ = QB Λ + gΨ ∗ Λ − gΛ ∗ Ψ .

(9.6.8)

The ∗ product is associative, and one can show by a contour argument that QB acts as a derivation, QB (Ψ1 ∗ Ψ2 ) = (QB Ψ1 ) ∗ Ψ2 + Ψ1 ∗ (QB Ψ2 ) .

(9.6.9)

From these it follows that the symmetry (9.6.8) closes. Using the cyclic property,

Ψ1 ∗ Ψ2 =

Ψ2 ∗ Ψ1 ,

(9.6.10)

313

9.6 String ﬁeld theory VΨ

VΨ

(a)

VΨ QB

(b)

VΨ

VΨ

Fig. 9.11. CFT representation of the string ﬁeld action: (a) the kinetic term; (b) the interaction. A z 2/3 conformal transformation is used to turn the usual semi-circle around each vertex operator into a wedge.

we can write the invariant action 1 2g S= Ψ ∗ QB Ψ + Ψ∗Ψ∗Ψ . (9.6.11) 2 3 This is also usefully written in terms of CFT, replacing Ψ with a half-disk with corresponding vertex operator VΨ , 1 2g

VΨ QB ·VΨ D2 +

V Ψ V Ψ V Ψ D 2 . (9.6.12) 2 3 The vertex operator positions and conformal frames are not explicitly denoted in eq. (9.6.12) but are as shown in ﬁgure 9.11. A z 2/3 mapping is used to ﬁt the three vertex operator half-disks into the disk. This action passes an important check. The path integral on the disk is nonzero only for insertions of total Qg = 3. The vertex operators for the physical states all have ghost number 1, so both terms in the action (9.6.12) are nonvanishing. Since the action (9.6.11) is invariant, we might expect that for the gauge ﬁeld it reduces to the non-Abelian Yang–Mills action, and this is the case. The Siegel gauge is b0 Ψ = 0. In the kinetic action, only the term c0 L0 from QB survives. Since in components L0 ∝ p2 + m2 , this is analogous to the familiar covariant gauge for electrodynamics. The propagator S=

∞ 0

dt exp(−tL0 ) = L−1 0

(9.6.13)

is simply the integral over strips of any length. The vertex is ﬁgure 9.11(b) with the wedges removed, joining the ends of three strips in a ‘Y.’ This has been shown to be equivalent to open string theory described as a sum over Riemann surfaces, including in particular a complete single cover of moduli space. On the one hand this is not surprising: the string ﬁeld theory has the spacetime gauge invariance (9.6.8), so that null states will

314

9 Higher order amplitudes

decouple from the S-matrix. We have argued that the decoupling of null states determines the form of the string amplitudes including the need for a complete cover of moduli space (else total derivative terms will give null state couplings). On the other hand moduli space is complicated and few explicit descriptions are known. The one produced by this string ﬁeld theory, in terms of a sum over Feynman graphs and an integral over strip lengths for each, was discovered by mathematicians only shortly before its construction through string ﬁeld theory.2 Let us point out a strange feature of this ﬁeld theory. We have stated that the string ﬁeld theory covers the entire moduli space, but we know from the annulus that this includes limits that produce closed string poles, even though there are no explicit closed string ﬁelds. The closed string poles come from strips going to zero, rather than inﬁnite, length.3 The precise statement is that the open string ﬁeld path integral (9.6.1) generates the amplitudes of open plus closed string theory (except that surfaces with no boundaries at all, to which open strings cannot couple, are not generated). In this respect it is as though closed strings were in some imprecise sense ‘bound states’ of open strings. For a theory with only closed strings we can try to repeat the above steps with a closed string ﬁeld, though it is somewhat more complicated. Already for the free action (9.6.3) the ghost number does not work out: we need an extra c ghost to make the matrix element between physical states nonvanishing. There is a remedy, which is to require the string ﬁeld Ψ and the gauge transformation Λ to be annihilated by b0 − ˜b0 and by ˜ 0 . The action L0 − L S0 =

1

Ψ|(c0 − ˜c0 )QB |Ψ

2

(9.6.14)

is then invariant under δΨ = QB Λ. Going on to the interacting theory there is no associative product analogous to ∗, and correspondingly no simple cover of moduli space. As in the previous section we need to add in higher vertices Vg,n , with more powers of the ﬁeld and also with internal handles, in order to cover moduli space. The resulting action, like the open string action, can be obtained from a symmetry principle using what is known as the Batalin–Vilkovisky formalism. Gauge ﬁxing then leads to the perturbation theory described in the previous section. 2

3

Another physically motivated description of moduli space comes from the light-cone gauge, where the moduli are the interaction times and the momenta p+ of the intermediate states, and also a twist angle for each intermediate closed string. For this reason, this form of string ﬁeld theory would not be suitable for our purposes of the previous section. We would need to add stubs to the vertex to eliminate the unexpected divergences, and then to add in closed string propagators and vertices to cover the omitted region of moduli space.

315

9.7 Large order behavior

The derivation of string theory from spacetime gauge symmetry is extremely elegant, particularly for the open string. The inclusion of closed strings in the open string ﬁeld theory is intriguing. However, it remains to be seen whether string ﬁeld theory is a useful tool beyond perturbation theory. It has not played a role in the recent nonperturbative progress based on spacetime supersymmetry, which will be described in chapter 14. It is quite possible that strings, or string ﬁelds, are simply not the correct degrees of freedom for a nonperturbative formulation of ‘string’ theory. 9.7

Large order behavior

There is an interesting result about the large order behavior of string perturbation theory. Let us start with ﬁeld theory. Consider the integral

∞

y2 y3 dy exp − − λ = λ−1 2! 3! −∞

∞

1 z2 z3 dz exp − 2 + λ 2! 3! −∞

,

(9.7.1)

where z = yλ. This integral diverges for λ = 0, but for now we can ignore that and calculate it as a formal expansion in powers of λ, ∞ (−λ)n ∞ n=0

6n n!

−∞

dy y 3n exp(−y 2 /2) = (2π)1/2

∞

λ2k C2k ,

(9.7.2)

k=0

with C2k =

2k Γ(3k + 1/2) . π 1/2 32k (2k)!

(9.7.3)

This integral is the zero-dimensional limit of a scalar ﬁeld theory, with a mass term and cubic coupling. The expansion in λ can be represented in terms of Feynman graphs, with propagator 1 and cubic vertex −λ. The constant C2k then simply counts the total number of vacuum graphs with 2k cubic vertices, or equivalently k + 1 loops, weighted by the appropriate symmetry factor. For example, the reader can check that there are two vacuum graphs with two vertices, with symmetry groups of order 8 and 5 1 = 18 + 12 . To get the large order behavior, we use 12, and indeed C2 = 24 n+1/2 e−n . Then Stirling’s approximation n! ≈ n C2k ≈ k k ≈ k! ,

(9.7.4)

where the approximation holds up to factors of the form k a and bk . One can also include a quartic term λ2 y 4 /4!, which makes the integral converge. The power of λ is determined by requiring that all terms in the action for z be proportional to λ−2 ; equivalently, all k + 1 loop vacuum graphs are weighted by λ2k . This additional term does not substantially change the result; the estimate (9.7.4) still holds to the given accuracy.

316

9 Higher order amplitudes

When considering ﬁeld theory, the propagator and vertices become more complicated. However, the dominant behavior at large orders usually comes from simple counting of graphs — the propagators and vertices only contribute factors of order bk , unless there is unusual IR or UV behavior. The series is of the form ∞

λ2k k!f2k ,

(9.7.5)

k=0

where f2k ≈ k a bk varies more slowly than k!. This diverges unless f2k has unexpected cancellations. For small λ the terms become smaller for a while and then grow as the k! dominates. The smallest terms in the sum are of order exp[−O(1/λ2 )] .

(9.7.6)

This is then an indication of the accuracy of perturbation theory as an asymptotic series. Indeed, eﬀects like instantons and dynamical symmetry breaking, which vanish to all orders of perturbation theory, are typically of this order. For example, the large order behavior of the series (9.7.2) is related to the ‘instanton’ (stationary point) at z = −2, with action 2/3λ2 . In string theory, the Feynman graph decomposition of moduli space makes it possible to establish the same bounds. It has been shown explicitly that the series is of the form (9.7.5) with f2k positive and bounded below by bk . Thus the series is divergent, and is not even Borel summable. Borel summation rewrites the series (9.7.5) as the Laplace transform of a more convergent series ∞ 0

dt exp(−t)

∞

(tλ2 )k f2k .

(9.7.7)

k=0

However, the positivity of the f2k implies that this new sum, even if it converges, has poles on the positive t-axis, so that the integral is ambiguous. Thus, we should expect that string theories will have interesting nonperturbative phenomena. This is not surprising because even in the low energy limit they reduce to nontrivial ﬁeld theories. The estimate (9.7.4) for closed string theory is a lower bound, which can be obtained even keeping only the tree-level cubic vertex V0,3 . Because the closed string vertex receives corrections from Riemann surfaces of all genus, one might wonder whether perturbation theory grows even more rapidly. One way to see that in fact it does is to recall that the open string ﬁeld theory contains all open plus closed string Riemann surfaces with at least one boundary. Let us consider the surfaces with one boundary, in the region where the boundary has shrunk to a point. This limit of open string moduli space is equivalent to the moduli space for closed surfaces

9.8 High energy and high temperature

317

with one vertex operator. One can show that this region is not vastly smaller than the full open string moduli space, so the same estimate holds, the perturbation series going as ∞

go2k O(k!) =

k=0

∞

gck O(k!) .

(9.7.8)

k=0

The genus-g term, proportional to gc2g , has a coeﬃcient of order (2g)!. This is larger than the g! characteristic of ﬁeld theory, which was also the lower bound obtained from V0,3 alone. Evidently the largest part of the large order amplitudes is contained within the vertices Vg,n and comes from the interior of moduli space. One then expects nonperturbative eﬀects of order exp[−O(1/gc )] ,

(9.7.9)

much larger at weak coupling than the exp[−O(1/gc2 )] characteristic of ﬁeld theory. This large order behavior was ﬁrst observed in some simpliﬁed models to be discussed in section 9.9. As we will see in chapter 13, these eﬀects are now known to be related to D-branes, at least in some supersymmetric string theories. 9.8

High energy and high temperature

In order to obtain a better understanding of string dynamics, one might look at the behavior in diﬀerent limits. In this section we consider various high energy limits of string theory. Hard scattering Consider again the hard scattering behavior, which we obtained from the Veneziano and Virasoro–Shapiro amplitudes by Stirling’s approximation. Go back to the integral form (6.6.5), 8πigc2 (2π)26δ 26 ( α

i ki )

C

d2 z4 |z4 |−α u/2−4 |1 − z4 |−α t/2−4 .

(9.8.1)

In the hard scattering region s → ∞ with t/s ﬁxed, the exponents are large and the integral is dominated by the stationary points of the exponent.4 The saddle point, for s, t, u 1, is given by u t ∂ u ln |z4 |2 + t ln |1 − z4 |2 = + = 0, ∂z4 z4 z4 − 1

4

(9.8.2)

As usual, the integral diverges at z4 → ∞ and is deﬁned by analytic continuation. The saddle point approximation can be justiﬁed for the continuation.

318

9 Higher order amplitudes

with solution z4 = −u/s; recall that s + t + u = O(1). The saddle point approximation then gives an amplitude S ≈ exp[−α (s ln s + t ln t + u ln u)/2] ,

(9.8.3)

as given in eq. (6.6.13). Let us also back up one step, from the integral over z4 to the path integral over X µ . This is dominated by the saddle point Xclµ (σ) = i

µ ki G (σ, σi ) ,

(9.8.4)

i

where the σi are the vertex operator positions. Recall that the Green’s function is proportional to α . At high energy the size of the scattering region grows linearly with energy, δX ≈ α k. In ﬁeld theory, higher energy hard scattering probes shorter distances, δX ≈ 1/k by the uncertainty principle. In string theory this is not the case beyond the string scale. It appears to be impossible in string perturbation theory to probe distances below this scale, giving rise to the idea that it is a limiting distance in spacetime. The eﬀective lower limit on compactiﬁcation R ≈ α1/2 also supports this. However, we will see in chapter 13 that recent advances in nonperturbative string theory reveal the existence of shorter length scales. At general orders in the string loop expansion, the dominant term in the amplitude will be

exp −

ki · kj G (σi , σj ) .

(9.8.5)

i<j

We need to ﬁnd the saddle points of the exponent, with respect to both the vertex operator positions and the moduli of the Riemann surface. This is like an electrostatics problem: the exponent is the energy of n point charges kiµ on a Riemann surface, where µ labels d diﬀerent electrostatic ﬁelds and µ = 0 has the opposite sign in the energy. The background charge density vanishes by momentum conservation, and the renormalized self-energy is negligible because ki2 remains on-shell and small. Remarkably, one can ﬁnd what is probably the dominant saddle point at each genus. Consider the Riemann surface (z1 − a1 )(z1 − a2 ) . (9.8.6) z2N = (z1 − a3 )(z1 − a4 ) Like the hypergeometric curve (9.2.9), it is a subspace of S2 × S2 . It can be regarded as N sheets glued together at branch cuts, and has genus N − 1. For the four-point amplitude, if the charges are placed at the branch cuts ai then the amplitude is stationary with respect to their positions — the surface is ‘vertical’ at these points, ∂z2 /∂z1 = ∞, and the symmetry z2 → exp(2πi/N)z2 requires the vertical component of the gradient to vanish. It remains to extremize with respect to the positions ai , but this

9.8 High energy and high temperature

319

is easy: the electric ﬁeld is the same as on the sphere with charges at z = ai , except that it is divided between N sheets. Thus the energy is N · N −2 = N −1 times that on the sphere. Three of the positions can be ﬁxed by M¨ obius invariance, for example to 0, 1, ∞, and the fourth is given by the same condition (9.8.2) as on the sphere. Thus at genus g, Sg ≈ exp[−α (s ln s + t ln t + u ln u)/2(g + 1)] .

(9.8.7)

With additional eﬀort one can include the Gaussian ﬂuctuations around the saddle, which determine the coeﬃcient of Sg . For the N-sheeted saddle points, Xclµ is simply divided by N = g + 1. There is a check that this result makes sense. In the small-angle region −t/s 1 it becomes exp[α t ln s/2(g + 1)] .

(9.8.8)

Now, t1/2 is the total momentum transferred from the incoming particle 1 to the ﬁnal particle 3. At small angles it is plausible that the genus-g scattering process can be decomposed into g + 1 successive scatterings. Each separate scattering has the same center of mass s, while the ti are constrained so that the total momentum transfer is t1/2 . The individual scatterings go as exp(α ti ln s/2). It is then easy to check that it is most 1/2 1/2 eﬃcient to take the momentum transfers to be equal, ti = t /(g + 1), leading to the result (9.8.8) for the product of the amplitudes. This is diﬀerent from particle scattering, where it is most eﬃcient to transfer momentum in a small number of hard scatterings. For excited string states, the exponential part of the vertex operator is the same as for the tachyon. The same saddle point then dominates, with Xclµ inserted in the vertex operators. Since this same saddle point dominates for all states and, up to the factor g + 1, all orders, there are linear relations between the amplitudes for diﬀerent states that hold for all genus and therefore plausibly for the exact amplitudes. In quantum ﬁeld theory, high energy scattering often reveals symmetries that are not obvious at low energy. It has been suggested that the relations that emerge here are of the same sort. It is not clear to what extent it makes sense to sum the series (9.8.7). It is interesting to contrast the result here with that in the previous section. In both cases it is the interior of moduli space that is producing the interesting physics (in contrast to the ﬁeld-theoretic behavior found near the boundary). However, the large order behavior came from the whole of moduli space, dominated in particular by its large volume. The high energy behavior at ﬁxed genus is dominated by a single point.

320

9 Higher order amplitudes Regge scattering

The analysis of the Regge limit can also be carried to higher orders. We will not do this, but will develop an interesting interpretation of the tree-level Regge physics. Regge scattering is less familiar to most physicists than hard scattering. It is a gentle probe of a highly boosted system. When a system is boosted, time dilation slows down its internal processes. One might then expect to see more of the virtual degrees of freedom as one goes to higher energies. Let us in particular consider the quantum ﬂuctuations of a single string, in particular the root mean size of the ground state. The mode expansion gives

0|[X 1 (σ) − x1 ]2 |0 =

∞ α

2n2

0|(αn α−n + ˜αn ˜α−n )|0

n=1 ∞

=α

1 . n n=1

(9.8.9)

This is divergent but has no direct physical signiﬁcance. A measurement on a time-scale δτ is sensitive only up to modes of frequency less than δτ−1 , essentially n α−1/2 < δτ−1 . The logarithmic divergence becomes α ln(α1/2 /δτ), and the size is the square root of this. This growth in the eﬀective size is directly visible in the Regge amplitude (6.6.12) at small t, Γ(−α t/4 − 1) s2+α t/2 Γ(α t/4 + 2)

s2 q 2 α ∼ exp − ln(α s) , t 2

(9.8.10)

where q 2 = −t is the momentum transfer. This is the gravitational amplitude s2 /t modiﬁed by a form-factor whose Fourier transform corresponds to an object of size (α ln α s)1/2 . From the point of view of one string the other has a boost of order α s, so the time resolution of its internal processes is inverse to this, in agreement with the earlier estimate for the size. The square root of a logarithm is a very slow function and ordinarily would be of little importance, but it has been argued that this behavior is very signiﬁcant for strings near black holes. In chapter 14 we will revisit, from a diﬀerent point of view, the idea that strings reveal their true nature when very highly boosted. High temperature A similar limit with interesting stringy behavior is high temperature. From the generic behavior (7.2.30) of the partition function at high weight h,

321

9.8 High energy and high temperature

and the relation m2 = 4(h − 1)/α , the density of string states n(m) satisﬁes ∞ 0

dm n(m) exp(−α πm2 ) ≈ exp(4π/) ,

(9.8.11)

implying n(m) ≈ exp(4πmα1/2 ). The thermal partition function of a single string is then ∞

dm exp(4πmα1/2 ) exp(−m/T ) .

(9.8.12)

0

Because the density of states grows exponentially, the partition function (9.8.12) diverges for temperatures greater than the Hagedorn temperature TH =

1 . 4πα1/2

(9.8.13)

One can understand the Hagedorn divergence as follows. The energy of a long string is proportional to its length. So also is its entropy, being extensive. At low temperature the energy dominates, but at high temperature the entropy dominates and it is favorable to produce strings of arbitrary length. In an ensemble of such strings the density will become large, and the interactions will be important near the transition. This divergence suggests a phase transition. Indeed, in the strong interaction there is a similar growth in the density of states of highly excited hadrons, and there is a phase transition from a low temperature hadron phase to a high temperature quark–gluon phase. It is interesting to wonder whether the Hagedorn divergence indicates some analogy to the quark–gluon structure in the fundamental string, but there is no clear answer. To conclude, let us discuss the partition function of free strings in somewhat more detail. The free energy density of a real scalar ﬁeld of mass m in dimension d is 2

dd−1 k ln[1 − exp(−ωk /T )] (2π)d−1 ∞ ∞ dt m2 t r2 −d/2 =− exp − . (2πt) − t 2 2T 2 t 0 r=1

F(T , m ) = T

(9.8.14)

In the second form, one can readily sum over the string spectrum. Following the derivation of the vacuum amplitudes (7.3.12) and (7.3.6), the free energy density in 26 ﬂat dimensions is F(T ) = −

R

∞ dτd¯τ r2 (4π 2 α τ2 )−13 |η(τ)|−48 exp − 2τ2 4πT 2 α τ2 r=1

.

(9.8.15)

322

9 Higher order amplitudes

The integral runs over the full region R:

τ1 ≤

1 , 2

|τ2 | > 0 .

(9.8.16)

The derivation from a modular-invariant path integral is given as exercise 9.7. There is a possible divergence as τ2 → 0; the integrand is largest when the real part of τ is zero, so let τ = iτ2 . From the modular transformation (7.2.44), the asymptotic behavior of the eta function in this limit is −1/2

η(iτ2 ) = η(i/τ2 )τ2 ≈ exp(−π/12τ2 ) as τ2 → 0 .

(9.8.17)

The τ integral thus converges for T < TH and diverges for T > TH . The following curious observation has been made. In ﬁeld theory the partition function is given by a Euclidean path integral, with time having period 1/T . The same is true in string theory (exercise 9.7). T -duality should then relate the high and low temperature behaviors, and in fact the Hagedorn temperature is half the self-dual value. Deﬁne ˜ ) = F(T ) + ρ0 , F(T

(9.8.18)

the thermal part (9.8.15) of the free energy plus the vacuum part (7.3.6). ˜ )/T , so T -duality implies The logarithm of the path integral is −F(T 1˜ T ˜ F(4TH2 /T ) . F(T ) = T 4TH2

(9.8.19)

This relates the partition function below the Hagedorn temperature to that above it. The Hagedorn divergence for T > TH is a reﬂection of the tachyon divergence in the low temperature theory, but a similar formal result holds for the supersymmetric string, which has no tachyonic divergence. If we take this relation seriously, we obtain 2 ˜ ) → T ρ0 F(T 4TH2

(9.8.20)

as T → ∞. If this is a measure of the number of high energy degrees of freedom, they are very few. A single quantum ﬁeld in d spacetime dimensions has a free energy of order T d , so it is as though the high energy limit of string theory looks like a ﬁeld theory in two spacetime dimensions. This is entirely unlike the exponentially growing spectrum below the Hagedorn transition, and may be a clue to the nature of the phase transition.

9.9 Low dimensions and noncritical strings 9.9

323

Low dimensions and noncritical strings

Field theories often simplify as the number of spacetime dimensions is reduced, and these low-dimensional theories have been useful models for understanding dynamics. We might try to do the same in string theory, but string theory requires a critical dimension. We could compactify, but this does not reduce the number of degrees of freedom: the string still oscillates in the compact directions. We have, however, encountered a consistent background in which the string actually does move in fewer dimensions. This is the linear dilaton background (3.7.18), Gµν (x) = ηµν ,

Φ(x) = Vµ xµ ,

Bµν (x) = 0 ,

1/2

(9.9.1a)

26 − D . (9.9.1b) 6α For D < 26 the dilaton gradient is spacelike and we have taken it to lie in the 1-direction. The eﬀective string coupling eΦ is a function of x1 , diverging at large x1 . This makes it hard to use string perturbation theory, but fortunately there is a slightly diﬀerent background for which perturbation theory is good. Inserting the background (9.9.1) into the tachyon action (6.6.16) leads to the linearized tachyon ﬁeld equation, Vµ = δµ1

4 T (x) = 0 . (9.9.2) α This is the condition for the exponential vertex operator to be Weylinvariant, with the linear dilaton energy-momentum tensor (2.5.1). The solutions are 2−D T (x) = exp(q · x) , (q − V )2 = . (9.9.3) 6α − ∂µ ∂µ T (x) + 2V µ ∂µ T (x) −

If we look for solutions which, like the dilaton, depend only on x1 , then

q1 = α ± =

26 − D 6α

1/2

±

2−D 6α

1/2

.

(9.9.4)

For D > 2 this is complex, so the tachyon oscillates, but for D ≤ 2 one gets a real exponential (times a linear function at the critical value D = 2). Of course, the tachyon ﬁeld equation will have nonlinear corrections. It can be shown that this does not aﬀect the qualitative form of the background, except that for rather subtle reasons only the solution α− leads to a nonsingular background. For D ≤ 2 the term ‘tachyon’ for the string ground state is actually a misnomer, because this ﬁeld is stable (exercise 9.8). Also for D ≤ 2, the real exponential T (x) = T0 exp(α− x1 ) prevents propagation to the region

324

tachyon background

weak coupling

9 Higher order amplitudes

Fig. 9.12. Spacetime with dilaton and tachyon backgrounds. The interaction strength grows toward the right but is cut oﬀ by an eﬀective potential from the tachyon background.

of large positive x1 where the coupling diverges. To see this, consider the world-sheet action

1 2 1/2 ab µ ν 1 1 Sσ = d σ g g η ∂ X ∂ X + α RV X + T exp(α X ) . µν a 1 0 − b 4πα M (9.9.5) At large x1 the exponential tachyon dominates the linear dilaton and the path integral is suppressed, corresponding to an eﬀective potential for the string due to the tachyon background. Physically one has an asymptotic region x1 → −∞, where the coupling goes to zero. Strings propagating toward positive x1 interact with one another and reﬂect oﬀ the potential and back to the asymptotic region. The maximum eﬀective coupling is set by the energy and the strength T0 of the potential. This picture, shown in ﬁgure 9.12, is a bit exotic, but in the interest of ﬁnding a solvable model it is worth pursuing. A model with rather similar physics, the static model, played a role in unraveling the nature of quantum ﬁeld theory. For D > 2 the physical eﬀect of the tachyon background is less clear, though it has been argued that it still acts as a barrier. However, the focus has been on D ≤ 2 for another reason, namely that these theories are actually solvable. Let us ﬁrst observe that when we quantize the string, the gauge symmetries take away two families of oscillators, so that in D ≤ 2 none will be left — there is only the center-of-mass motion, a single

9.9 Low dimensions and noncritical strings

325

scalar.5 Nevertheless, to solve the string theory directly is still quite hard. First one has to evaluate the world-sheet path integral with action (9.9.5). This is no longer a free theory, because of the exponential tachyon ﬁeld. It is known as the Liouville ﬁeld theory. It is classically integrable, and with substantial eﬀort some quantum amplitudes have been determined. However, to solve the string theory one would still need to integrate the result at each genus over the moduli space of Riemann surfaces; we have seen that this is a complicated space. Remarkably, the full amplitude has been determined, to all orders, by very diﬀerent means. To describe this, we must ﬁrst introduce another idea. Noncritical strings Consider a statistical mechanical system deﬁned by a sum over twodimensional surfaces in a d-dimensional Euclidean space. Let us also suppose that the surfaces have some internal degree of freedom that can be interpreted as a measure of length. The natural ﬁelds to describe the system are the embedding X µ (σ) and a metric gab (σ). However, there is no particular reason for the length to drop out of the problem — that is, for the theory to be Weyl-invariant. This is therefore called the noncritical string. The lowest-dimensional action (or Boltzmann weight, since we are thinking of the path integral in the context of statistical mechanics) for this system would then be the Polyakov action with an additional worldsheet cosmological constant term, which in the critical string is forbidden by Weyl invariance, 1 S= 4πα

M

d2 σ g 1/2 g ab ∂a X µ ∂b Xµ + µ .

(9.9.6)

Without Weyl invariance, we can use diﬀ invariance to ﬁx the metric partially, but one degree of freedom remains: gab (σ) = exp(2φ(σ)) gˆ ab (σ) ,

(9.9.7)

where gˆ ab is a gauge choice and the scale φ of the metric is still pathintegrated. The Faddeev–Popov determinant for this gauge-ﬁxing is the same as in eq. (3.4.19), the Weyl-ﬁxing contribution to the latter being trivial. Inserting the metric (9.9.7), for which the curvature is

ˆ 2φ , g 1/2 R = gˆ 1/2 Rˆ − 2∇

(9.9.8)

and dropping a ﬁeld-independent term, the action plus Faddeev–Popov 5

At special momenta one ﬁnds additional physical states. These can be interpreted as unbroken symmetries of string ﬁeld theory, and likely play a role in the solvability of the theory.

326

9 Higher order amplitudes

determinant becomes

13α ab ˆ gˆ ∂a φ∂b φ + Rφ . 3 M (9.9.9) Note that the Faddeev–Popov determinant has provided a kinetic term for φ. Comparing this with the dilaton–tachyon action (9.9.5) we see a marked resemblance. If we let d = D − 1 but interpret the new ﬁeld φ as x1 , and if we rename gab → gˆ ab in the action (9.9.5) (since it can be fully gauge-ﬁxed), then all the same terms appear. Some coeﬃcients are a bit diﬀerent — there is no rescaling of φ that makes them all agree — but in fact we claim that these are the same ﬁeld theory. The point is that the critical string path integral (9.9.5) is deﬁned using gˆ ab as the metric, and the noncritical string (9.9.9) using exp(2φ)gˆ ab . A careful treatment of the Weyl transformation of the measure then converts one to the other. One can understand this quite easily by noting that only the full metric (9.9.7) appears in the deﬁnition of the noncritical string, so nothing can change under

S=

1 4πα

d2 σ gˆ 1/2 gˆ ab ∂a X µ ∂b Xµ + µ exp 2φ +

gˆ ab (σ) → exp(2ω(σ)) gˆ ab (σ) , φ(σ) → φ(σ) − ω(σ) .

(9.9.10a) (9.9.10b)

The noncritical string thus acquires a Weyl invariance, and this is enough to ﬁx the various coeﬃcients, other than the free parameter µ. It may seem odd that Weyl-invariant and noninvariant theories can be equivalent. The point is that in ﬁeld theory we can always add extra ‘fake’ gauge symmetries if we also add extra ﬁelds that we just use to ﬁx the gauge. A gauge symmetry after all is just a redundancy of description, and we can always adopt a more redundant description. For the symmetry (9.9.10) it is clear that we can just cancel the Weyl symmetry against the ﬁeld φ, but for general string backgrounds (for example ﬂat space, where the embedding coordinates are Weyl-invariant) this is not possible and the Weyl invariance is ‘real.’ In any case we are now nearing the punch line. One can approximate the noncritical string by a sum of triangulated surfaces, the number of edges between two points being a measure of length. These triangulated surfaces resemble large Feynman diagrams, and in fact one can ﬁnd a quantum system, the matrix model, for which this is the diagram expansion. The ﬁelds of the matrix model are N × N matrices. Taking N → ∞ and adjusting other couplings, there is (at least for D ≤ 2) a critical point dominated by large smooth surfaces, so we recover the noncritical string. Moreover, the matrix model is solvable (again for D ≤ 2) by diagonalizing the matrices. In the end one has a rather simple system (noninteracting

Exercises

327

fermions moving in the space of eigenvalues) which by retracing the above steps allows the calculation of string amplitudes to all genus. Because these methods are so far aﬁeld from the rest of the book, we will have to leave the details to the references. In any case they have not yet been useful at D > 2 or in supersymmetric theories. (However, we will see in chapter 14 that a somewhat diﬀerent theory of large-N matrices has recently been proposed as a nonperturbative string theory.) We should mention, though, at least one important lesson from these models. This is the (2g)! behavior of perturbation theory, which was seen ﬁrst in the explicit matrix model solution and then argued to hold in all string theories. In the matrix model the corresponding exp[O(−1/gc )] eﬀect involves tunneling of the fermions in the space of eigenvalues. It is not clear whether this can be related to the exp[O(−1/gc )] eﬀects in higher-dimensional string theories. Exercises 9.1 (a) Use a conformal transformation to show that the OPE (9.1.2) and expectation value (9.1.3) imply the inner product (9.1.29). ˆ j (k) , verify the second (b) By taking matrix elements in the basis |V equality in equation (9.1.28). 9.2 (a) Write the four-point amplitude on the torus in operator form, parallel to eq. (9.1.17) for the sphere, but now involving a trace. ˆ 1 = {QB , X}, then (b) Show that if one of the vertex operators is null, V the trace is equal to total derivatives with respect to the vertex operator positions plus total derivatives with respect to τ and ¯τ. 9.3 (a) For the torus with four vertex operators, use the result of part (a) of the previous problem to study the unitarity in the limit that w1 , w2 , and w3 − w4 are ﬁxed and w3 and 2πτ − w3 approach i∞. This is two handles degenerating, and unitarity relates the amplitude to the product of two tree-level four-point amplitudes. (b) Repeat in particular the demonstration that the sum over the full ˆ and that over the physical Hilbert space are equal. Show Hilbert space H that this requires cancellation between intermediate states of diﬀerent ghost numbers. This cancellation is a stringy generalization of the cancellation between Faddeev–Popov ghosts and longitudinal gauge bosons in gauge theory. 9.4 Verify the assertion in eq. (9.3.7) about the region covered by the Feynman graph of ﬁgure 9.6(a), and ﬁnd the other two regions covered by this graph. 9.5 (a) For the terms (9.6.7) in the expansion of the string ﬁeld, write out explicitly the gauge transformation (9.6.2) and the action (9.6.3).

328

9 Higher order amplitudes

(b) Eliminate auxiliary ﬁelds and ﬁelds that can be completely gauged away, and show that the resulting gauge transformation and action for Aµ have the expected form. 9.6 For f2k = k a bk , show that the smallest term in the series (9.7.5) is of the order of exp(−C/g 2 ). Work to suﬃcient accuracy to ﬁnd C. For the series (9.7.2), compare with the instanton action. 9.7 In quantum theory, the Euclidean path integral with Euclidean time ˜ )/T ]. having period T −1 gives the thermal partition function exp[−F(T Show that for the string path integral this agrees with the result (9.8.15). The one subtlety is an interplay between the region of integration and the sum over windings (Reference: McClain & Roth, (1987).) 9.8 For the tachyon wave equation (9.9.2) in the linear dilaton background (9.9.1), show that any ﬁnite energy initial conﬁguration has stable (oscillatory) evolution provided that D ≤ 2. 9.9 Study the physical state conditions (either in OCQ or BRST form) for the graviton level of the closed string in the D = 2 linear dilaton background. Let the momenta take complex values. You should ﬁnd that the physical spectrum is trivial at generic momenta but that at two momenta there are nontrivial physical states. 9.10 Derive eq. 9.8.14. Hint: in the second line introduce an integral of k such that m2 becomes m2 + k2 . Then integrate over t (the result is in terms of elementary functions. Finally, the sum on r is the series expansion of the log in the ﬁrst line.

Appendix A A short course on path integrals

Path integrals are a powerful means of representing quantum theories, especially on surfaces of nontrivial topology. The introduction we present here is similar to that which the reader will ﬁnd in any modern ﬁeld theory text. We include it in order to emphasize certain ideas that we will need, such as the relation between the path integral and Hilbert space formalisms and the use of operator equations inside the path integral. A.1

Bosonic ﬁelds

Consider ﬁrst a quantum mechanics problem, one degree of freedom with ˆ p, q), ˆ where Hamiltonian H(ˆ ˆ pˆ ] = i¯h . [q,

(A.1.1)

Throughout the appendix operators are indicated by hats. A basic quantity of interest is the amplitude to evolve from one q eigenstate to another in a time T , ˆ /¯h)|qi .

qf | exp(−iHT

(A.1.2)

In ﬁeld theory it is generally convenient to use the Heisenberg representation, where operators have the time dependence ˆ = exp(iHt/¯ ˆ h) A ˆ exp(−iHt/¯ ˆ h) . A(t)

(A.1.3)

ˆ The state |q, t is an eigenstate of q(t), ˆ q(t)|q, t = |q, t q .

(A.1.4)

In terms of the t = 0 Schr¨ odinger eigenstates this is ˆ h)|q . |q, t = exp(iHt/¯ In this notation the transition amplitude (A.1.2) is qf , T |qi , 0 . 329

(A.1.5)

330

Appendix A

qf

...

T

t2 t1 0

qi

Fig. A.1. Transition amplitude broken down into time steps. The dashed line suggests a piecewise linear path, which is one of many ways to deﬁne path integration.

By inserting a complete set of states, we can write the transition amplitude as a coherent sum over all states q through which the system might pass at some intermediate time t:

qf , T |qi , 0 =

dq qf , T |q, t

q, t|qi , 0 .

(A.1.6)

Divide the time interval further into N steps as in ﬁgure A.1, tm = m ,

= T /N ,

(A.1.7)

and corresponding to each intermediate time insert a complete set of states

qf , T |qi , 0 =

dqN−1 . . . dq1

N−1

qm+1 , tm+1 |qm , tm .

(A.1.8)

m=0

Translate back to Schr¨ odinger formalism and introduce an integral over intermediate momenta, ˆ h)|qm

qm+1 , tm+1 |qm , tm = qm+1 | exp(−iH/¯

=

ˆ h)|qm . dpm qm+1 |pm

pm | exp(−iH/¯

(A.1.9)

ˆ with all pˆ to the left and all qˆ to By commuting, we can always write H the right, so that ˆ p, q)|q ˆ m = H(pm , qm ) pm |qm .

pm |H(ˆ

(A.1.10)

Use this to evaluate the matrix element in eq. (A.1.9). From expanding the exponential there will be terms of order 2 and higher that have qˆ to the

331

A.1 Bosonic ﬁelds left of pˆ ; drop these. Then to this order qm+1 , tm+1 |qm , tm becomes

dpm exp[−iH(pm , qm )/¯ h] qm+1 |pm

pm |qm

=

!

i

dpm exp − H(pm , qm ) − pm (qm+1 − qm ) + O(2 ) 2π¯ h ¯ h

Thus

qf , T |qi , 0 ≈

→

.

(A.1.11)

dpN−1 dp0 dp1 dqN−1 . . . dq1 2π¯h 2π¯h 2π¯h

"

i N−1 × exp − H(pm , qm ) − pm (qm+1 − qm ) ¯h m=0 ! T i

"

dt [ p˙ q − H(p, q) ] . (A.1.12) h¯ 0 In the last line we have taken a formal → 0 limit, so the integral runs over all paths p(t), q(t) with given q(T ) and q(0). This is the Hamiltonian path integral (or functional integral), the sum over phase space paths weighted by exp(iS /¯h) with S the action in Hamiltonian form. Let us integrate over p(t), making ﬁrst the approximation that the integral is dominated by the point of stationary phase, 0=

[dpdq] exp

∂ ∂ [ p˙ q − H(p, q) ] = q˙ − H(p, q) . ∂p ∂p

(A.1.13)

Solving for p in terms of q and q˙ and recalling that L = p˙ q − H, the stationary phase approximation gives

qf , T |qi , 0 =

T i

[dq] exp

¯h

dt L(q, q˙) .

(A.1.14)

0

ˆ = 1 pˆ 2 + V (q), ˆ so that the Often the momentum integral is Gaussian, H 2 stationary phase approximation is exact up to a normalization that can be absorbed in the measure [dq]. In fact, we can do this generally: because the relation (A.1.13) between p and q˙ is local, the corrections to the stationary phase approximation are local and so can be absorbed into the deﬁnition of [dq] without spoiling the essential property that [dq] is a product over independent integrals at each intermediate time. The result (A.1.14) is the Lagrangian path integral, the integral over all paths in conﬁguration space weighted by exp(iS /¯h), where now S is the Lagrangian action. We have kept ¯h so as to discuss the classical limit. As ¯h → 0, the integral is dominated by the paths of stationary phase, δS /δq(t) = 0, which is indeed the classical equation of motion. Henceforth, h¯ = 1. The derivation above clearly generalizes to a multicomponent q i . One can always write a quantum ﬁeld theory in terms of an inﬁnite number of qs, either by making the spatial coordinate x discrete and taking a limit,

332

Appendix A

or by expanding the x-dependence in terms of Fourier modes. Thus the above derivation of the path integral representation applies to bosonic ﬁeld theory as well. The derivation of the path integral formula treated the continuum limit → 0 rather heuristically. Renormalization theory is the study of this limiting process, and is rather well understood. The limit exists provided the Lagrangian satisﬁes certain conditions (renormalizability) and provided the coeﬃcients in the Lagrangian (bare couplings) are taken to vary appropriately as the limit is taken. The result is independent of various choices that must be made, including the operator ordering in the Hamiltonian, the precise deﬁnition of the measure, and the diﬀerences between regulators (the limit of piecewise linear paths suggested by the ﬁgure versus a frequency cutoﬀ on the Fourier modes, for example).1 Being local eﬀects these can all be absorbed into the bare couplings. Actually, we will evaluate explicitly only a few Gaussian path integrals. We use the path integral primarily to derive some general results, such as operator equations and sewing relations. Relation to the Hilbert space formalism Starting from the path integral one may cut it open to recover the Hilbert space of the theory. In equation (A.1.6), if one writes each of the amplitudes as a path integral one ﬁnds

T

qf ,T

[dq]q ,0 exp i

=

i

dq

dtL 0 qf ,T

T

[dq]q,t exp i

dtL t

q,t

t

[dq]q ,0 exp i i

dtL

. (A.1.15)

0

That is, the path integral on [0, T ] breaks up into separate path integrals on [0, t] and [t, T ], plus an ordinary integral over q(t). The boundary conditions on paths are denoted by super/subscripts. Note the restriction that T ≥ t ≥ 0. One can also replace the intermediate integral with a sum over any complete set of states. This cutting (or, inversely, sewing) principle is particularly useful on the surfaces of nontrivial topology that arise in string theory. There one may cut the surface open along many diﬀerent closed curves; diﬀerent choices give diﬀerent Hilbert space representations of a given amplitude. This is known as world-sheet duality. Consider now a path integral with the additional insertion of a factor 1

As another example, one can freely rescale ﬁelds. The Jacobian is formally a product of separate Jacobians at each point; its logarithm is therefore a sum of a local expression at each point. Regulated, this is just a shift of the bare couplings.

333

A.1 Bosonic ﬁelds of q(t), where 0 < t < T . Then, using the cutting relation (A.1.15),

qf ,T

[dq]q ,0 exp(iS )q(t) = i

=

dq qf , T |q, t q q, t|qi , 0

ˆ dq qf , T |q(t)|q, t q, t|qi , 0

ˆ = qf , T |q(t)|q i , 0 .

(A.1.16)

ˆ Thus, q(t) in the functional integral translates into q(t) in the matrix element. For a product of two insertions q(t)q(t ) (which is equal to q(t )q(t), since these are simply variables of integration), where both t and t are in the range [0, T ], one ﬁnds by the same method that

qf ,T

ˆ q(t ˆ )]|qi , 0 . [dq]q ,0 exp(iS )q(t)q(t ) = qf , T |T[q(t)

(A.1.17)

Here, T denotes the time-ordered product ˆ B(t ˆ ) + θ(t − t)B(t ˆ )A(t). ˆ ˆ B(t ˆ )] = θ(t − t )A(t) T[A(t)

(A.1.18)

i

Because of the way the path integral is built out of successive inﬁnitesimal time slices, two or more insertions in the path integral will always correspond to the time-ordered product of operators in the matrix element. The equation of motion is δS /δq(t) = 0. Let us see how this works inside the path integral. Consider this equation in a path integral with other insertions F,

qf ,T

[dq]q ,0 i

δ [dq]q ,0 exp(iS ) F i δq(t) qf ,T δF . (A.1.19) = i [dq]q ,0 exp(iS ) i δq(t)

δS exp(iS ) F = −i δq(t)

qf ,T

We have integrated by parts in the second line.2 If none of the ﬁelds in F are at time t, the right-hand side vanishes and the equation of motion holds in the path integral. We therefore say that the equation of motion holds as an operator equation. This is a technique that we will use extensively: an operator equation is one which holds when inserted into any path integral that has no other ﬁelds coincident with those involved in the equation. Translating to Hilbert space language, we can use the other insertions F to set up arbitrary initial and ﬁnal states, so this corresponds 2

The functional derivative is deﬁned as δ q(t ) = δ(t − t ) . δq(t) It can be regarded as the limit of the ordinary derivative 1 ∂ , ∂qm

m = t/ .

In particular, one can integrate a functional derivative by parts, as for an ordinary derivative.

334

Appendix A

to the ordinary notion that an operator equation is one that should hold for general matrix elements. When additional insertions are present at t, operator equations inside the path integral hold only up to additional delta-function source terms. For example, let F = q(t ) . . . , where t is near t and the other ﬁelds in ‘. . .’ are far away. Then eq. (A.1.19) becomes -

q(t )

δS ... δq(t)

.

= i δ(t − t ) . . .

.

(A.1.20)

This is an example of the Schwinger–Dyson equation, the equation of motion for the expectation value of a product of ﬁelds. The same result written as an operator equation is q(t )

δS = iδ(t − t ) . δq(t)

(A.1.21)

As an explicit example, take the harmonic oscillator, 1 S = 2

dt (˙ q2 − ω2q2) ,

δS = −¨ q (t) − ω 2 q(t) . δq(t) Then the operator equations we have derived are 2 ∂

(A.1.22a) (A.1.22b)

+ ω 2 q(t) = 0

(A.1.23)

+ ω 2 q(t)q(t ) = −iδ(t − t ) .

(A.1.24)

∂t2 and 2 ∂

∂t2

In Hilbert space language these become 2 ∂ 2 ∂

+ω

∂t2

2

ˆ =0, q(t)

ˆ q(t ˆ )] = −iδ(t − t ) . + ω 2 T[q(t)

(A.1.25a) ∂t2 The ﬁrst is the Heisenberg equation for the ﬁeld operator, while the delta function in the second comes from the time derivative acting on the step functions in the time-ordered product, 2 ∂

∂t2

+ω

2

#

ˆ q(t ˆ )] = T T [q(t)

$

¨ˆ + ω 2 q(t) ˙ˆ q(t)] ˆ ˆ ˆ ) + δ(t − t )[q(t), q(t) q(t

= 0 − iδ(t − t ) .

(A.1.26)

335

A.1 Bosonic ﬁelds Euclidean path integrals In Schr¨ odinger language, the time-ordered product (A.1.17) is ˆ ˆ > − t< )] qˆ exp[−iHt ˆ < ]|qi ,

qf | exp[−iH(T − t> )] qˆ exp[−iH(t

(A.1.27)

where t> is the greater of t and t , and t< is the lesser. By the nature of the path integral, which builds amplitudes from successive time steps, the time diﬀerences multiplying H are always positive in amplitudes such as (A.1.27): one only encounters exp(−iH∆t) for positive ∆t. Now continue analytically by uniformly rotating all times in the complex plane, t → e−iθ t for any phase 0 ≤ θ ≤ π. The real part of the exponent is −E sin θ∆t, where E is the energy of the intermediate state. Because the Hamiltonian is bounded below for a stable system, the sum over intermediate states is safely convergent and no singularities are encountered — the continued amplitude is well deﬁned. Taking the times to be purely imaginary, t = −iu for real u, deﬁnes the Euclidean amplitudes, as opposed to the Minkowski amplitudes studied above:

qf , u|T

ˆ a ) |qi , 0 E q(u

(A.1.28)

a

is deﬁned to be the continuation of

qf , t|T

ˆ a ) |qi , 0

q(t

(A.1.29)

a

to t = −iu. One can repeat the derivation of the path integral representation for the Euclidean case, but the result is extremely simple: just replace t in the action with −iu,

qf , U|T

ˆ a ) |qi , 0 E = q(u

a

U

qf ,U

[dq]q ,0 exp i

0

du L(q, i∂u q)

q(ua ) .

a

(A.1.30) The weight in the Euclidean path integral is conventionally deﬁned to be exp(−SE ), so the Euclidean action contains a minus sign,

SE =

du LE (q, ∂u q) = −

du L(q, i∂u q) .

(A.1.31)

Euclidean path integrals are usually the more well-deﬁned, because the exponential of the action is damped rather than oscillatory. The Minkowski matrix elements can then be obtained by analytic continuation. In the text our main concern is the Euclidean path integral, which we use to represent the string S-matrix. One is often interested in traces, such as ˆ exp(−Ei U) , (A.1.32) Tr exp(−HU) = i

336

Appendix A

ˆ and the Ei are the correwhere the sum runs over all eigenvectors of H sponding eigenvalues. In the q basis the trace becomes

ˆ dq q| exp(−HU)|q

=

=

dq q, U|q, 0 E

dq

q,U

[dq]q,0 exp(−SE )

[dq]P exp(−SE ) .

=

(A.1.33)

In the last line, [dq]P denotes the integral over all periodic paths on [0, U]. An interesting variation on this is the integral over all antiperiodic paths:

[dq]A exp(−SE ) =

=

=

q,U

[dq]−q,0 exp(−SE )

dq

dq q, U| −q, 0 E ˆ 0

dq q, U|R|q, E

ˆ Rˆ . = Tr exp(−HU)

(A.1.34)

Here Rˆ is the reﬂection operator, taking q to −q. We see that twisting the boundary conditions — for example, replacing periodic with antiperiodic — is equivalent to inserting an operator into the trace. Since Rˆ 2 = 1, its eigenvalues are ±1. If we take the average of the periodic and antiperiodic ˆ which projects onto the Rˆ = +1 space: path integrals, this inserts 12 (1 + R), 1 exp(−Ei U) . (periodic + antiperiodic) = 2 R =+1

(A.1.35)

i

In the same way, 12 (periodic − antiperiodic) gives the trace on the Rˆ = −1 space. ˆ is a Hermitean operator, A minor but sometimes confusing point: if A ˆ ˆ A ˆ exp(−Hu) ˆ satisﬁes the Euclidean Heisenberg operator A(u) = exp(Hu) ˆ ˆ † = A(−u) A(u)

(A.1.36)

and so is not in general Hermitean. For this reason it is useful to deﬁne a Euclidean adjoint, † ˆ ˆ . A(u) = A(−u)

(A.1.37)

A Hermitean operator in the Schr¨ odinger picture becomes Euclidean self-adjoint in the Euclidean Heisenberg picture.

337

A.1 Bosonic ﬁelds Diagrams and determinants

The diagrammatic expansion is easily obtained from the path integral. This subject is well treated in modern ﬁeld theory texts. We will only derive here a few results used in the text. For notational clarity we switch to ﬁeld theory, replacing q(t) with a real scalar ﬁeld φ(t, x). Begin with the free action 1 2

dd x φ(x)∆φ(x) ,

(A.1.38)

where ∆ is some diﬀerential operator. Consider the path integral with a source J(x), taking the Euclidean case to be speciﬁc,

.

d

exp i

Z[J] = =

d x J(x)φ(x)

1 [dφ] exp − 2

d

d

d x φ(x)∆φ(x) + i

d x J(x)φ(x) . (A.1.39)

This is the generating functional for Euclidean Green’s functions. Taking functional derivatives with respect to J(x) pulls down factors of iφ(x) into the integral. Shifting φ to φ + i∆−1 J completes the square, leaving

1 Z[J] = Z[0] exp − dd x dd y J(x)∆−1 (x, y)J(y) 2 1 δ2 d d −1 = Z[0] exp d x d y ∆ (x, y) 2 δφ(x)δφ(y)

× exp i

d x J(x)φ(x) d

. (A.1.40) φ=0

In the second line, each φ derivative becomes iJ. Any functional F of φ can be written F[φ] =

[dJ] exp i

dd x J(x)φ(x) f[J] ,

(A.1.41)

where f[J] is the (functional) Fourier transform of F. By linearity one obtains

1

F = Z[0] exp 2

δ2 d x d y ∆ (x, y) F[φ] . (A.1.42) δφ(x)δφ(y) φ=0 d

d

−1

Expanding the exponential generates the diagrammatic expansion, all ways of contracting pairs of ﬁelds in F. Also, when F is a product of factors, F1 F2 . . . , we can separate δ δ δ = + + ... , δφ(x) δφ1 (x) δφ2 (x)

(A.1.43)

338

Appendix A

where δ/δφi acts only on Fi . The self-contractions are then the terms δ2 δφi (x)δφi (y)

(no sum on i) .

(A.1.44)

Consider now the overall factor Z[0]. We assume that ∆ is self-adjoint and so we expand φ(x) in a complete set of eigenfunctions Φi (x): φ(x) =

φi Φi (x) ,

(A.1.45)

∆Φi (x) = λi Φi (x) ,

(A.1.46a)

i

where

dd x Φi (x)Φj (x) = δij . Deﬁne the path integral measure by [dφ] =

(A.1.46b)

(A.1.47)

dφi .

i

Then

1 =

i

=

1 dφi exp − φ2i λi 2

2π 1/2

λi

i

=

∆ det 2π

−1/2

.

(A.1.48)

The product of eigenvalues must be suitably regulated and renormalized. The value of 1 is usually given simply as (det ∆)−1/2 , because a rescaling of ∆ is equivalent to a rescaling of φ, the Jacobian for which can be absorbed into the bare couplings. (In free ﬁeld theory, the only ‘coupling’ that would be renormalized would be a c-number constant in the Lagrangian density; we will see this appearing in the example.) All of the above discussion extends readily to multicomponent q i or i φ . One case of interest is a complex scalar ﬁeld. One can break this up into two real scalars, φ = 2−1/2 (φ1 + iφ2 ), and the measure is [dφ dφ∗ ] ≡ [dφ1 dφ2 ]. For complex ﬁelds, (A.1.48) becomes

∗

[dφ dφ ] exp −

dd x φ∗ ∆φ = (det ∆)−1 .

(A.1.49)

One similarly ﬁnds for two real ﬁelds

[dφ1 dφ2 ] exp i

dd x φ1 ∆φ2 = (det ∆)−1 .

(A.1.50)

339

A.1 Bosonic ﬁelds This is a functional version of the delta-function integral ∞ −∞

dx

∞ −∞

dy exp(iλxy) = 2π

∞ −∞

dx δ(λx) =

2π . λ

(A.1.51)

An example As an example, we study the Euclidean transition amplitude for the harmonic oscillator,

qf , U|qi , 0 E =

qf ,U

[dq]q ,0 exp(−SE ) ,

(A.1.52a)

i

1 U

du (∂u q)2 + ω 2 q 2 + Sct , (A.1.52b) 2 0 where Sct is a cutoﬀ-dependent piece to be determined later. To evaluate this, separate out the ‘classical’ part of q,

SE =

q(u) = qcl (u) + q (u) ,

(A.1.53)

where −¨ qcl (u) + ω 2 qcl (u) = 0 , qcl (0) = qi , qcl (U) = qf .

(A.1.54a) (A.1.54b)

The boundary conditions on q are then q (0) = q (U) = 0

(A.1.55)

SE = Scl (qi , qf ) + S + Sct .

(A.1.56)

and the action separates

Here Scl (qi , qf ) =

1 2

U 0

du (∂u qcl )2 + ω 2 qcl2

(qi2 + qf2 ) cosh ωU − 2qi qf =ω , 2 sinh ωU S =

1 2

U 0

du (∂u q )2 + ω 2 q 2 .

The amplitude becomes

qf , U|qi , 0 = exp[−Scl (qi , qf ) − Sct ]

0,U

[dq ]0,0 exp(−S ) .

(A.1.57a) (A.1.57b)

(A.1.58)

Expand in a complete set of eigenfunctions of ∆ = −∂u2 + ω 2 subject to

340

Appendix A

the boundary condition (A.1.55),

2 1/2 jπu j 2π2 fj (u) = sin + ω2 , (A.1.59a) , λj = U U U2 ∞ 2 2 ∆ j π + ω2U 2 = det . (A.1.59b) 2 2π 2πU j=1 The inﬁnite product diverges. The Pauli–Villars regulator is a simple way to deﬁne it. Divide by the amplitude for a regulator oscillator of very high frequency Ω, det

∆ 2π

regulate

→

∞ 2 2 j π + ω2U 2 j=1

j 2 π 2 + Ω2 U 2

(A.1.60)

.

For j small, the terms in (A.1.60) are the same as those in (A.1.59b) up to a multiplicative constant, while for j large the product converges. The value of the product (A.1.60) is Ω sinh ωU . (A.1.61) ω sinh ΩU This follows from a standard inﬁnite product representation for the sinh. One can guess the result (A.1.61) by noting that it has the right poles and zeros in U, and the right value as U goes to zero. Keeping only terms that are nonvanishing as Ω → ∞,

1/2

1

qf , U|qi , 0 → exp −Scl (qi , qf ) + (ΩU − ln Ω) − Sct . 2 (A.1.62) To get a ﬁnite answer as Ω → ∞, we need ﬁrst to include a term 12 Ω in the Lagrangian Lct , canceling the linear divergence in (A.1.62). That is, there is a linearly divergent bare coupling for the operator ‘1’. It may seem strange that we need to renormalize in a quantum mechanics problem, but the power counting is completely uniform with quantum ﬁeld theory. The logarithmic divergence is a wavefunction renormalization. To determine the ﬁnite normalization, compare the path integral expression (A.1.62) to the canonical result as U → ∞. The U → ∞ amplitude is easily evaluated by inserting a complete set of states, since only the ground state |0

contributes in the limit: ω 2 sinh ωU

qf , U|qi , 0 → qf |0

0|qi exp(−E0 U)

ω 1/2 ω exp − (qf2 + qi2 + U) . (A.1.63) = π 2 On the other hand, the path integral expression (A.1.62) becomes ω 2 1 1/2 2

qf , U|qi , 0 → ω exp − (qf + qi + U) + (ΩU − ln Ω) − Sct . (A.1.64) 2 2

A.2 Fermionic ﬁelds Thus 1 Sct = 2 and

qf , U|qi , 0 =

U 0

du Ω −

ω 2π sinh ωU

341

1 Ω ln 2 π

(A.1.65)

exp(−Scl ) ,

(A.1.66)

1/2

which is the correct result. A.2

Fermionic ﬁelds

Consider the basic fermionic quantum system, two states | ↑ and | ↓ with ˆ raising operator χˆ and lowering operator ψ: ˆ ↓ = 0 , χˆ | ↓ = | ↑ , ψ| ˆ ↑ = | ↓ , χˆ | ↑ = 0 , ψ| ˆ χˆ } = 1 , ψˆ 2 = χˆ 2 = 0 . {ψ,

(A.2.1a) (A.2.1b) (A.2.1c)

Amplitudes will be expressed as path integrals over the classical analogs ˆ These are anticommuting c-numbers or Grassmann variables, of χˆ and ψ. elements χm and ψm of a Grassmann algebra {χm , ψn } = {χm , χn } = {ψm , ψn } = 0 .

(A.2.2)

There is a natural notion of integration over Grassmann variables, the Berezin integral. Notice that if θ is a Grassmann variable then θ2 = 0, and so the most general function of θ is of the form a + θb. Deﬁne

dθ = 0 ,

dθ θ = −

θ dθ = 1 .

(A.2.3)

Integrals of more general expressions are determined by requiring the integral to be linear and by deﬁning dθ to anticommute with all Grassmann variables. The signiﬁcance of the deﬁnition (A.2.3) is that the integral of a total derivative is zero: d dθ (a + θb) = dθ b = 0 . (A.2.4) dθ For multiple Grassmann variables the integral is

dθ1 . . . dθn f(θ) = c ,

(A.2.5)

where c is the coeﬃcient of the highest term in the Taylor expansion of f, f(θ) = . . . + θn θn−1 . . . θ1 c .

(A.2.6)

Looking back on the discussion of the bosonic path integral, one sees that all of the properties of ordinary integrals that were used are also true for

342

Appendix A

the Berezin integral: linearity, invariance (except for a sign) under change of order of integration, and integration by parts.3 In order to follow the bosonic discussion as closely as possible, it is ˆ useful to deﬁne states that are formally eigenstates of ψ: |ψ = | ↓ + | ↑ ψ , ˆ ψ|ψ

= |ψ ψ .

(A.2.7a) (A.2.7b)

ψ|ψ = ψ − ψ .

(A.2.8)

Deﬁne ψ| to satisfy The right-hand side is the Grassmann version of the Dirac delta function. It has the same properties as the ordinary delta function, (ψ − ψ )f(ψ) = (ψ − ψ )f(ψ ) and

dψ (ψ − ψ )f(ψ) = f(ψ )

(A.2.9)

(A.2.10)

for arbitrary f(ψ). The state ψ| is a left eigenstate with opposite sign,

ψ|ψˆ = −ψ ψ| ,

(A.2.11)

as one veriﬁes by taking the inner product of both sides with a general state |ψ . Note also that

|ψ dψ ψ| = 1 .

(A.2.12)

One can verify this by taking the matrix element between arbitrary states

ψ | and |ψ and evaluating the Grassmann integral. The delta function has an integral representation −

dχ exp[χ(ψ − ψ)] = ψ − ψ = ψ|ψ .

(A.2.13)

By expanding the inner product (A.2.8) in powers of ψ one obtains

ψ| ↓ = ψ ,

ψ| ↑ = −1 ,

(A.2.14)

and therefore

ψ|ˆχ|ψ = ψ| ↑

=−

3

dχ χ exp[χ(ψ − ψ)] .

(A.2.15)

It is not useful to think of dθ as an integral in any ordinary sense: it is simply a linear mapping from functions of θ to the complex numbers. Since it shares so many of the properties of ordinary integrals, the integral notation is a useful mnemonic.

343

A.2 Fermionic ﬁelds ˆ one obtains Using the further property that |ψ is an eigenstate of ψ, ˆ = − ψ|f(ˆχ, ψ)|ψ

dχ f(χ, ψ ) exp[χ(ψ − ψ)] .

(A.2.16)

This holds for any function f with the prescription that it is written with the ψˆ on the right. We can now emulate the bosonic discussion, using the completeness relation (A.2.12) to join together successive time steps, with the result

ψf , T |ψi , 0 ≈ −

dχ0 dψ1 dχ1 . . . dψN−1 dχN−1 × exp

→

N−1

−iH(χm , ψm ) − χm (ψm+1 − ψm )

m=0

! T

[dχdψ] exp 0

˙ − iH(χ, ψ) dt −χψ

"

.

(A.2.17)

The inconvenient sign in the integral representation has been oﬀset by reversing the order of Grassmann integrations. A typical fermionic Hamilˆ = mˆχψ. ˆ The corresponding Lagrangian is L = iχψ ˙ − H. For tonian is H such ﬁrst order Lagrangians the Hamiltonian and Lagrangian formulations coincide, and the result (A.2.17) is also the Lagrangian path integral. There are two useful inner products for this system, (A) : (B) :

↑| ↑ = ↓| ↓ = 1 ,

↑| ↓ = ↓| ↑ = 1 ,

↑| ↓ = ↓| ↑ = 0 ,

↑| ↑ = ↓| ↓ = 0 .

(A.2.18a) (A.2.18b)

Inner product (A) is positive deﬁnite and gives ψˆ † = χˆ . It is the inner product for an ordinary physical oscillator. Inner product (B) is not positive, and it makes ψˆ and χˆ Hermitean. It is the appropriate inner product for the Faddeev–Popov ghost system, as discussed in chapter 4. The derivation of the path integral was independent of the choice of inner product, but the relations (A.2.14) determine the expansion of ψ| in basis states in terms of the inner product. All of the discussion of the bosonic case applies to the fermionic path integral as well. The derivation is readily extended to multiple Fermi oscillators and to ﬁeld theory. Renormalization theory works in the same way as for the bosonic path integral. Fermionic path integrals again satisfy cutting relations, which simply amount to taking one of the iterated integrals (A.2.17) and saving it for the end. Insertion of ψ(t) and χ(t) in the path integral again gives the corresponding operator in the matrix element, and multiple insertions are again time-ordered. The path integral insertion ψ(t)ψ(t ) = −ψ(t )ψ(t)

(A.2.19)

344

Appendix A

becomes the operator ˆ ψ(t ˆ )] ≡ θ(t − t )ψ(t) ˆ ψ(t ˆ ) − θ(t − t)ψ(t ˆ )ψ(t) ˆ T[ψ(t) .

(A.2.20)

Note that a minus sign appears in the deﬁnition of time-ordering. The equations of motion and the Euclidean continuation work as in the bosonic ˆ case, but there is one important change. Consider a general operator A, ˆ = |j Aji , A|i

(A.2.21)

ˆ where i, j run over ↓ and ↑. Then by expanding A|ψ

in terms of the basis states one ﬁnds

ˆ dψ ψ|A|ψ

= A↓↓ − A↑↑

ˆ

ˆ , = Tr (−1)F A

(A.2.22)

where the fermion number Fˆ has been deﬁned to be even for | ↓ and odd for | ↑ . In particular,

ˆ

ˆ dψ ψ, U|ψ, 0 = Tr (−1)F exp(−HU) .

(A.2.23)

In contrast to the bosonic results (A.1.33) and (A.1.34), the periodic path ˆ integral gives the trace weighted by (−1)F and the antiperiodic path integral gives the simple trace. Feynman diagrams are as in the bosonic case, but the determinants work out diﬀerently. Consider

1 =

[dψ dχ] exp

d

d x χ∆ψ

,

(A.2.24)

where we have again switched to the notation of ﬁeld theory. Expand ψ in eigenfunctions of ∆, and χ in eigenfunctions of ∆T , ψ(x) =

ψi Ψi (x) ,

i

χ(x) =

χi Υi (x) ,

∆Ψi (x) = λi Ψi (x) ,

∆T Υi (x) = λi Υi (x) ,

dd x Υi (x)Ψj (x) = δij . Deﬁne [dψdχ] =

i dψi dχi .

1 =

(A.2.25a)

i

Then

(A.2.25b) (A.2.25c)

dψi dχi exp(λi χi ψi )

i

=

λi

i

= det ∆ .

(A.2.26)

345

Exercises Similarly, one ﬁnds for a single ﬁeld ψ that

[dψ] exp

dd x ψ∆ψ = (det ∆)1/2 ,

(A.2.27)

where ∆ must be antisymmetric. These fermionic path integrals are just the inverse of the corresponding bosonic integrals. For ﬁnite-dimensional integrals we have to be careful about 2πs, ∞

∞

1 dx dy exp(2πiλxy) = = λ −∞ −∞

−1

dψ

dχ exp(λχψ)

. (A.2.28)

Curiously, we can be more cavalier for functional determinants, where these can be eliminated by rescaling the ﬁelds. Bosonic path integrals actually give the absolute values of determinants, while fermionic path integrals give the determinant with a sign that depends on the precise order of Grassmann factors in the measure. We will ﬁx the sign by hand.4 If ∆ has zero eigenvalues, the path integral vanishes in the fermionic case and diverges in the bosonic case. When this occurs there is always a good reason, as we will see in practice, and the integrals of interest will have appropriate insertions to give a ﬁnite result. A useful result about Berezin integrals: from the deﬁnition it follows that if θ = θa then dθ = a−1 dθ. That is, the Jacobian is the inverse of the bosonic case. More generally, with bosonic variables xi and Grassmann variables θj , under a general change of variables to xi (x, θ), θj (x, θ) the inﬁnitesimal transformation of the measure has an extra minus sign, m

n

m

n

δ(d x d θ) = d x d θ

i

∂ ∂ δxi − δθj ∂xi ∂θj j

.

(A.2.29)

Exercises A.1 (a) Evaluate the path integral (A.1.33) for the harmonic oscillator by expanding [dq]P in a set of periodic eigenfunctions (not by inserting the result already obtained for the transition function into the left-hand side). Compare with an explicit evaluation of the sum over states (A.1.32). (b) Do the same for the antiperiodic path integral (A.1.34). A.2 Consider two harmonic oscillators of the same frequency, and form the complex coordinate q = 2−1/2 (q1 + iq2 ). Evaluate the following path 4

A ﬁne point: when the world-sheet theory is parity-asymmetric, even the phase of the path integral is ambiguous. The point is that ∆ is not Hermitian in this case, because the Euclidean adjoint includes a time-reversal (which is the same as a parity ﬂip plus Euclidean rotation). One must instead take the determinant of the Hermitian operator ∆† ∆, which determines that of ∆ only up to a phase. We will encounter this in section 10.7, for example, where we will have to check that there is a consistent deﬁnition of the phase.

346

Appendix A

integral on [0, U] by using a suitable mode expansion:

[dq dq ∗ ]θ exp(−SE ) ,

where the integral is to run over all paths such that q(U) = q(0)eiθ . Again compare with the answer obtained by summing over states: it is useful here to form complex combinations of oscillators, a1 ± ia2 . This is a more exotic version of exercise A.1, and arises in various string theories such as orbifolds. A.3 Consider a free scalar ﬁeld X in two dimensions, with action

1 d2 σ ∂1 X∂1 X + ∂2 X∂2 X + m2 X 2 . 4πα Evaluate the path integral with both coordinates periodic,

S=

0 ≤ σ1 ≤ 2π ,

0 ≤ σ2 ≤ T .

[Expanding the spatial dependence in Fourier modes, the answer takes the form of an inﬁnite product of harmonic oscillator traces.] A.4 Consider a free particle on a circle, H=−

1 ∂2 . 2 ∂φ2

By means of path integrals, evaluate φf , T |φi , 0 . Be careful: there are topologically inequivalent paths, distinguished by the number of times they wind around the circle. A.5 The fermionic version of exercise A.1: evaluate the fermionic path integral (A.2.23) and the corresponding antiperiodic path integral by using a complete set of periodic or antiperiodic functions, and compare with the sum over states.

References

The literature on the subjects covered in these two volumes easily exceeds 5000 papers, and probably approaches twice that number. To represent accurately the contributions of all who have worked in these many areas is an undertaking beyond the scope of this book. Rather, I have tried to assemble a list that is short enough to be useful in developing speciﬁc subjects and in giving points of entry to the literature. The resulting list includes many review articles, plus papers from the original literature where I felt they were needed to supplement the treatment in the text. Papers listed as ‘e-print: hep-th/yymmnnn’ are available electronically at the Los Alamos physics e-print archive, http://xxx.lanl.gov/abs/hep-th/yymmnnn . For a list of corrections and other useful information see http://www.itp.ucsb.edu/˜joep/bigbook.html . General references The reader may ﬁnd the two volume set by Green, Schwarz, & Witten (1987) (henceforth GSW) a useful supplement. The mix of covariant and light-cone methods used in that book, and the advanced topics emphasized, are somewhat complementary to the approach taken here. The book by Polyakov (1987) emphasizes conformal ﬁeld theory and the path integral approach to strings, but the focus is less on uniﬁed string theories and more on string theories of the strong interaction and on the statistical mechanics of random surfaces. Some other lectures on string theory are Peskin (1987), L¨ ust & Theisen (1989), Alvarez-Gaum´e & Vazquez-Mozo (1992), D’Hoker (1993), Ooguri & Yin (1997), and Kiritsis (1997). 347

348

References A brief history

There have been three periods of intense activity in the history of string theory, covering roughly the years 1968–73, 1984–9, and 1995 to the present. The ﬁrst of these began with the Veneziano amplitude as a phenomenological proposal for the S-matrix of the strong interaction. This led to dual resonance models, where the Veneziano formula was generalized and systematized. Further discoveries were the connection of dual models to strings, the critical dimension (26), superconformal invariance, and the existence of strings with critical dimension 10. This ﬁrst period of activity was ended by the discovery of QCD as the correct theory of the strong interaction, but those who had worked on string theory were left with the belief that they had discovered a rich physical theory that should have some application in nature. During the succeeding ‘quiet’ period, a small group of theorists developed string theory as a uniﬁed theory of gravity and gauge interactions. String theories with spacetime supersymmetry were discovered during this period, and the argument made that superstrings were a ﬁnite theory of quantum gravity. Three discoveries in 1984 established that string theory was also a strong candidate for the uniﬁcation of the Standard Model. These discoveries were the existence of anomaly-free chiral theories in ten dimensions, the heterotic string, and Calabi–Yau compactifaction. The ensuing period of activity was largely devoted to working out their consequences. The main diﬃculty that remained, and still remains, was the enormous number of potential vacua and an insuﬃcient understanding of the dynamics that chooses among them. The latest period of activity is marked by new dynamical tools. Again, many important results date from the preceding quiet period, but their importance and their relation to one another took time to emerge. The key steps were the systematic analysis of the consequences of supersymmetry in gauge theories and then in string theory. This led to many new discoveries: weak–strong duality in string theory, the uniﬁcation of all string theories as limits of a single theory, a limit in which a new, eleventh, dimension appears, the signiﬁcance of D-branes, and new understanding of the quantum mechanics of black holes. The developments from 1984 on are covered in the literature and review articles below and in volume two, but the earlier period is somewhat neglected. Many of the key papers up to 1985 can be found in Schwarz (1985), which also includes summaries of the history. Chapters 1 and 2 of the reference section of GSW give a more detailed history of the dual model period, with extensive references. The review by Scherk (1975) also covers this period.

References

349

Chapter 1 There are now many treatments of the Standard Model; Quigg (1983) gives a survey and Cheng & Li (1991) a more detailed treatment, to name just two. Weinberg (1979) discusses the UV problem of quantum gravity in ﬁeld theory. References for the various unifying ideas will be given in the appropriate chapters. The classical motion of the string and light-cone quantization are discussed in Goddard, Goldstone, Rebbi, & Thorn (1973) and in the reviews by Rebbi (1974) and Scherk (1975). The ﬁrst two of these include a careful treatment of Lorentz invariance. The heuristic argument that we have given is based on Brink & Nielsen (1973). Weinberg (1964) discusses the couplings of massless vector and tensor particles. Chapter 2 In addition to the general references, Friedan, Martinec, & Shenker (1986) covers many of the subjects in this chapter. Belavin, Polyakov, & Zamolodchikov (1984) is a classic reference on CFT, but directed at the advanced topics to be treated in chapter 15. Ginsparg (1990) is a review of CFT; Gatto (1973) and Fradkin & Palchik (1978) treat conformal invariance in four-dimensional quantum ﬁeld theory; although this is much smaller than the two-dimensional symmetry, many of the same ideas apply. Chapter 3 Additional references on the Polyakov path integral are Polyakov (1981), Friedan (1982), and Alvarez (1983). The Faddeev–Popov procedure is treated in all modern ﬁeld theory textbooks. The basic steps follow the original reference Faddeev & Popov (1967). The book by Birrell & Davies (1982) discusses Weyl anomalies in various dimensions (though it uses the common terminology conformal anomaly instead). Gravitational anomalies are discussed in Alvarez-Gaum´e & Witten (1983). An LSZ-like derivation of the vertex operators in light-cone gauge is in Mandelstam (1973); see also chapter 11 of GSW (1987). For calculation of beta functions in general nonlinear sigma models see Alvarez-Gaum´e, Freedman, & Mukhi (1981) and Friedan (1985). A general reference on strings in background ﬁelds is Callan, Friedan, Martinec, & Perry (1985); reviews include Callan & Thorlacius (1989) and Tseytlin (1989). The linear dilaton theory is discussed as a CFT by Chodos & Thorn (1974) and as a string background by Myers (1987).

350

References Chapter 4

Classic proofs of the OCQ no-ghost theorem are Brower (1972) and Goddard & Thorn (1972). The basic reference on quantization with constraints is Dirac (1950); a review is Henneaux (1985). The BRST symmetry in gauge theory is discussed in modern ﬁeld theory texts. The structure of its cohomology is discussed in Kugo & Ojima (1979). The BRST operator for ﬁeld theory is derived in Kato & Ogawa (1983); our proof of the noghost theorem is loosely patterned on the one in that paper. Other proofs in the BRST context are Freeman & Olive (1986), Frenkel, Garland, & Zuckerman (1986), and Thorn (1987). Chapter 5 D’Hoker & Phong (1988) is an extensive review of the subjects in this and subsequent chapters. The measure on moduli space is obtained in Moore & Nelson (1986) and D’Hoker & Phong (1986); see also Alvarez (1983). The representation of the measure in terms of the transition functions is discussed in Martinec (1987) and Alvarez-Gaum´e, Gomez, Moore, & Vafa (1988). General properties of the amplitude are also discussed in Friedan, Martinec, & Shenker (1986) and Mansﬁeld (1987). Chapter 6 Some of the physics of the tree-level amplitudes is discussed in Veneziano (1974). The appearance of gauge and gravitational interactions in the longdistance limit is developed in Neveu & Scherk (1972), Yoneya (1974), and Scherk & Schwarz (1974). Constraints on open string gauge groups are discussed in Marcus & Sagnotti (1982) and Schwarz (1982). For more on operator methods see GSW and the dual model references Schwarz (1973) and Mandelstam (1974). The relation of closed to open string tree-level amplitudes is given in Kawai, Lewellen, & Tye (1986). Chapter 7 The two volumes by Mumford (1983) are a beautiful introduction to theta functions; the one-loop functions are discussed in the ﬁrst half of volume one. Constraints from modular invariance on general conformal ﬁeld theories are in Cardy (1986). The zero-point energy in quantum ﬁeld theory is discussed in Coleman & Weinberg (1973). The cosmological constant problem is reviewed in Weinberg (1989).

References

351

Cancellation of divergences for open superstring gauge group S O(32) is discussed in Green & Schwarz (1985). The analysis using the boundary state formalism is in Callan, Lovelace, Nappi, & Yost (1987) and Polchinski & Cai (1988). Chapter 8 Kaluza–Klein theory is reviewed in Duﬀ, Nilsson, & Pope (1986) and in the reprint volume Appelquist, Chodos, & Freund (1987). Giveon, Porrati, & Rabinovici (1994) review toroidal compactiﬁcation and T -duality. Polchinski (1997) reviews T -duality of open and unoriented strings, and Dbranes. Further references on toroidal compactiﬁcation are Narain (1986) and Narain, Sarmadi, & Witten (1987). Dixon, Harvey, Vafa, & Witten (1985) is an introduction to orbifolds. Twisted state vertex amplitudes are obtained in Dixon, Friedan, Martinec & Shenker (1987) and Hamidi & Vafa (1987). More extensive orbifold references are given with chapter 16. The space of c = 1 CFTs is described in Dijkgraaf, Verlinde, & Verlinde (1988) and Ginsparg (1988) and in the review Ginsparg (1990). Chapter 9 Unitarity of tree-level amplitudes is discussed in the dual model reviews Schwarz (1973) and Mandelstam (1974) (these discussions are in OCQ, whereas section 9.1 gives the BRST version of the argument). One-loop amplitudes were originally constructed by imposing unitarity on the treelevel amplitudes; see Scherk (1975) for a review. The moduli space of higher genus Riemann surfaces is reviewed in Martinec (1987) and D’Hoker & Phong (1988). Some more details of the Schottky construction can be found in Mandelstam (1974) and references therein. Sewing of CFTs has been discussed by many authors. The treatment here is based on Vafa (1987), Polchinski (1988), Sonoda (1988a,b,1989) and LeClair, Peskin, & Preitschopf (1989a,b). The extension to the open string is in Lewellen (1992). The graphical decomposition of moduli space and the construction of minimal area metrics are reviewed in Zwiebach (1993). A discussion of the combinatorics of unitarity in quantum ﬁeld perturbation theory can be found in ’t Hooft & Veltman (1973). The Fischler–Susskind mechanism is reviewed in Polchinski (1988). For a diﬀerent approach to string unitarity, based on showing the equivalence between the light-cone and Polyakov amplitudes, see D’Hoker & Giddings (1987). The light-cone formalism is discussed in Mandelstam (1973) and GSW, chapter 11.

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Thorn, C. B. (1987). A detailed study of the physical state conditions in covariantly quantized string theories. Nuclear Physics, B286, 61. Thorn, C. B. (1989). String ﬁeld theory. Physics Reports, 175, 1. Tseytlin, A. A. (1989). Sigma model approach to string theory. International Journal of Modern Physics, A4, 1257. Vafa, C. (1987). Conformal ﬁeld theory and punctured surfaces. Physics Letters, B199, 195. Veneziano, G. (1974). An introduction to dual models of strong interactions and their physical motivations. Physics Reports, 9, 199. Weinberg, S. (1964). Photons and gravitons in S matrix theory. Physical Review, 135, B1049. Weinberg, S. (1979). Ultraviolet divergences in quantum theories of gravitation. In General Relativity. An Einstein Centenary Survey, eds. S. W. Hawking & W. Israel, pp. 790–831. Cambridge: Cambridge University Press. Weinberg, S. (1989). The cosmological constant problem. Reviews of Modern Physics, 61, 1. Witten, E. (1986). Noncommutative geometry and string ﬁeld theory. Nuclear Physics, B268, 253. Yoneya, T. (1974). Connection of dual models to electrodynamics and gravidynamics. Progress of Theoretical Physics, 51, 1907. Zwiebach, B. (1993). Closed string ﬁeld theory: quantum action and the Batalin-Vilkovisky master equation. Nuclear Physics, B390, 33. E-print hep-th/9206084.

Glossary

(0, 2) compactiﬁcation a generic heterotic string vacuum having d = 4, N = 1 supersymmetry. The world-sheet CFT has N = 2 right-moving superconformal invariance. (2, 0) theory in current usage, this refers to a family of nontrivial ﬁxed point theories with tensionless strings and d = 6 (2, 0) supersymmetry. These arise on coincident M5-branes and IIA NS5-branes, and on the IIB theory at an A–D–E singularity. (2, 2) compactiﬁcation one of a special subset of d = 4, N = 1 heterotic string vacua, which includes the Calabi–Yau compactiﬁcations. The world-sheet CFT has both right-moving and left-moving N = 2 superconformal invariance. In the type II string theories, these CFTs give vacua with d = 4, N = 2 supersymmetry. A–D–E singularity a singularity of a four-(real)-dimensional complex manifold, resulting from the collapse of one or more two-spheres to zero volume. The terminology A–D–E refers to the Dynkin diagrams of the simply-laced Lie algebras, which describe the intersection numbers of the collapsed spheres. Abelian diﬀerential a globally deﬁned holomorphic (1,0)-form on a Riemann surface. abstruse identity one of a set of quartic theta function identities due to Jacobi, it implies the degeneracy of bosons and fermions in GSO-projected string theories as required by supersymmetry. aﬃne Lie algebra see current algebra. anomaly the violation of a classical symmetry by quantum eﬀects. A gravitational anomaly is an anomaly in coordinate invariance. A global anomaly is an anomaly in a large symmetry transformation (one not continuously connected to the identity). anomaly polynomial a formal (d + 2)-form in d-dimensions, which encodes the gauge and gravitational anomalies.

359

360

Glossary

asymptotically locally Euclidean (ALE) space a space which at long distance approaches ﬂat Euclidean space identiﬁed under a discrete group. This is the geometry in the neighborhood of an orbifold ﬁxed point (or blown-up ﬁxed point). Atiyah–Drinfeld–Hitchin–Manin (ADHM) construction a method for the construction of all Yang–Mills ﬁeld conﬁgurations having self-dual ﬁeld strength. auxiliary ﬁeld a nonpropagating ﬁeld, one whose ﬁeld equation is algebraic rather than diﬀerential. In many supersymmetric theories, the transformations can be simpliﬁed by introducing such ﬁelds. axion a Goldstone boson associated with spontaneously broken PQ symmetry. The model-independent axion appears in every perturbative string theory, and is closely related to the graviton and dilaton. bc CFT a free CFT of anticommuting ﬁelds with an action of ﬁrst order in derivatives. There is a family of such CFTs, parameterized by the weight hb = 1 − hc . For hb = 2 this CFT describes the Faddeev–Popov ghosts associated with conformal invariance. βγ CFT a free CFT of commuting ﬁelds with an action of ﬁrst order in derivatives. There is a family of such CFTs, parameterized by the weight hβ = 1 − hγ . For hβ = 32 this CFT describes the Faddeev–Popov ghosts associated with superconformal invariance. Batalin–Vilkovisky formalism an extension of the BRST formalism, for quantizing more general theories with constraints. This has been useful in string ﬁeld theory. Becchi–Rouet–Stora–Tyutin (BRST) invariance a nilpotent symmetry of Faddeev–Popov gauge-ﬁxed theories, which encodes the information contained in the original gauge symmetry. Beltrami diﬀerential the derivative with respect to the moduli of the complex structure of a Riemann surface. Berezin integration a linear operation taking functions of Grassmann variables to complex numbers, with many of the key properties of ordinary integration. beta function 1. in quantum ﬁeld theory, the derivative of the eﬀective strength of an interaction with respect to length scale; 2. a special function involving a ratio of gamma functions, which appears in the Veneziano amplitude. Betti numbers the number of nontrivial p-forms in de Rham cohomology, denoted Bp . black hole entropy a quantity S proportional to the area of the horizon of a black hole, S = 2πA/κ2 . This has the properties of a thermodynamic entropy: it is nondecreasing in classical general relativity, and the sum of the black hole entropy and the ordinary entropy is nondecreasing even with the inclusion of Hawking radiation. To ﬁnd a statistical mechanical

Glossary

361

derivation of this entropy has been a major goal, partly realized in recent work. black hole evaporation the emission of thermal (Hawking) radiation by a black hole, due to pair production near the horizon. black hole information paradox a conﬂict between quantum mechanics and general relativity. Information falling into a black hole is lost and does not reappear when the black hole evaporates; this is inconsistent with ordinary quantum mechanical evolution. It apparently requires either a signiﬁcant modiﬁcation of quantum mechanics, or a signiﬁcant breakdown of the usual understanding of locality. black p-brane a p-dimensional extended object with an event horizon: a space that is translationally invariant in p directions and has a black hole geometry in the remaining directions. blow up to deform a singular manifold into a smooth manifold. Bogomolnyi–Prasad–Sommerﬁeld (BPS) state a state that is invariant under a nontrivial subalgebra of the full supersymmetry algebra. Such states always carry conserved charges, and the supersymmetry algebra determines the mass of the state exactly in terms of its charges. BPS states lie in smaller supersymmetry representations than non-BPS states, so-called short representations. When there are short representations of diﬀerent sizes, one also distinguishes ultrashort representations, which are the smallest possible (generally their dimension is the square root of the non-BPS dimension). Borel summation a method of deﬁning the sum of a divergent series. This has been used as a means of studying nonperturbative eﬀects in ﬁeld and string theories, but it should be understood that most nonperturbative eﬀects are not usefully studied in terms of the perturbation series. Born–Infeld action a generalization of the usual gauge ﬁeld action which is nonpolynomial in the gauge ﬁeld strength. This was originally proposed as a possible short-distance modiﬁcation of electromagnetism. It arises as the low energy eﬀective action of the gauge ﬁelds on D-branes. bosonization the exact equivalence of a theory of fermionic ﬁelds and a theory of bosonic ﬁelds, possible in two dimensions. The boson is a fermion– antifermion pair; the fermion is a coherent state of bosons. c-map a method for constructing the hypermultiplet moduli space of a type II string theory compactiﬁed on a Calabi–Yau three-fold from the vector multiplet moduli space of the other type II theory on the same three-fold. c-theorem the existence, in unitary CFTs in two dimensions, of a positive quantity c that is monotonically nonincreasing with increasing length scale, and which at ﬁxed points is stationary and equal to the central charge. This is a strong constraint on the global form of the renormalization group ﬂow; no simple analog seems to exist in higher dimensions. Also known as the Zamolodchikov c-theorem.

362

Glossary

C PT symmetry the combined operation of parity-reversal, time-reversal, and charge conjugation, which is a symmetry of all Lorentz-invariant local quantum ﬁeld theories. Calabi–Yau manifold a K¨ahler manifold with vanishing ﬁrst Chern class. A Calabi–Yau n-fold has 2n real = n complex coordinates. Yau’s theorem guarantees the existence of a Ricci-ﬂat metric of SU(n) holonomy. canceled propagator argument a general principle implying, under broad conditions, the vanishing of surface terms on the moduli space of Riemann surfaces and therefore the decoupling of unphysical states in string amplitudes. Such amplitudes are deﬁned by analytic continuation from a regime where the integrand falls rapidly at the boundary and the surface term is identically zero; its continuation is therefore also identically zero. Casimir energy a shift in the ground state energy of a quantum ﬁeld theory due to boundary conditions on the ﬁelds. center-of-mass mode the zeroth spatial Fourier component of a quantum ﬁeld. central charge an operator (which might be a constant) that appears on the right-hand side of a Lie algebra and commutes with all operators in the algebra. Prominent examples include the constant term in the Virasoro algebra and the charges appearing on the right-hand sides of many supersymmetry algebras. Chan–Paton degrees of freedom degrees of freedom localized at the endpoints of open strings. These are now interpreted as designating the D-brane on which the string ends. Chan–Paton factor the vertex operator factor for the state of the Chan–Paton degrees of freedom. Chern–Simons term a term in the action which involves p-form potentials as well as ﬁeld strengths. Such a term is gauge-invariant as a consequence of the Bianchi identity and/or the modiﬁcation of the p-form gauge transformation. These terms usually have a close connection to topology and to anomalies. chiral 1. acting in a parity asymmetric fashion; see chiral multiplet, chiral symmetry, chiral theory, chirality, extended chiral algebra; 2. invariant under part of the supersymmetry algebra; see chiral ﬁeld, chiral multiplet, chiral primary, chiral ring. chiral ﬁeld in supersymmetry, a local operator that is invariant under part of the algebra: the operator analog of a BPS state. chiral multiplet the multiplet of d = 4, N = 1 supersymmetry with two real scalars. The quarks and leptons are contained in such multiplets. This multiplet is connected to both senses of chiral: it contains a fermion with chiral couplings, and the integral of the associated superﬁeld is invariant under half of the supersymmetry algebra. chiral primary in an N = 2 SCFT, a primary ﬁeld that is also annihilated by one of the rigid supersymmetries G± −1/2 .

Glossary

363

chiral ring the closed OPE algebra of chiral ﬁelds. chiral symmetry a symmetry whose action on spinor ﬁelds is not paritysymmetric. chiral theory a gauge theory in which the gauge couplings are not paritysymmetric. chirality in d = 2k dimensions, the eigenvalue of the operator Γd which anticommutes with the Γµ . This eigenvalue distinguishes the two Weyl representations, which are related to one another by parity. Christoﬀel connection in general relativity, the connection that is constructed from the metric. critical behavior the behavior of a quantum ﬁeld theory at an IR ﬁxed point with massless ﬁelds, and the approach to this behavior. closed string a string with the topology of a circle. cocycle in a vertex operator, an operator-valued phase factor which multiplies the creation–annihilation normal ordered exponential. This in needed in some cases in order to give the operator the correct commuting or anticommuting property. coeﬃcient functions the position-dependent coeﬃcients of local operators appearing in the expansion of an operator product. cohomology in any vector space with a nilpotent operator Q (one such that Q2 = 0), the kernel of Q modulo the image of Q. That is, the space of closed states (those annihilated by Q) with the exact states (those of the form Qψ) deﬁned to be equivalent to zero. De Rham cohomology is the cohomology of the exterior derivative d acting on diﬀerential forms. On a complex manifold, Dolbeault cohomology is the cohomology of ∂ and ∂¯ (the (1, 0) and (0, 1) parts of d) on (p, q)-forms. Homology is the cohomology of the boundary operator. BRST cohomology is the cohomology of the BRST operator, and deﬁnes the physical space of a gauge-invariant theory. Coleman–Weinberg formula the expression for the vacuum energy density of a free quantum ﬁeld, from the renormalized sum of the zero-point energies of its modes. collapsing cycle a cycle whose volume vanishes in a limit, usually giving rise to a singular manifold. collective coordinate in quantizing a soliton or other extended object, the degrees of freedom corresponding to its position or conﬁguration. compact CFT a CFT in which the number of states with energy less than any given value is ﬁnite. This is deﬁned by analogy with the spectrum of a diﬀerential operator on a compact space. compactiﬁcation scale the characteristic mass scale of states whose wavefunctions have a nontrivial dependence on the compact dimensions. compactify to consider a ﬁeld theory or string theory in a spacetime, some of whose spatial dimensions are compact.

364

Glossary

complex manifold a manifold with an assigned system of complex coordinates, modulo holomorphic reparameterizations of these coordinates. complex structure an equivalence class of complex coordinates. A given diﬀerentiable manifold may have many inequivalent complex structures. complex structure moduli the moduli that parameterize the inequivalent complex structures on a manifold. In compactiﬁcation on a Calabi–Yau 3-fold, these are associated with (2, 1)-forms. conformal block in CFT, the contribution of a single conformal family to a sum over states. conformal bootstrap the partially successful program to construct all CFTs by using only symmetry and consistency conditions. conformal family the set of states obtained by acting on a highest weight state with Virasoro raising generators in all inequivalent ways; or, the corresponding set of local operators. A degenerate conformal family contains null states, which are orthogonal to all states in the family. conformal ﬁeld theory (CFT) a conformally invariant quantum ﬁeld theory. conformal gauge a choice of coordinates in two dimensions, such that the metric is proportional to the unit metric. conformal Killing vector a globally deﬁned inﬁnitesimal diﬀ×Weyl transformation that leaves the metric invariant. conformal transformation a mapping of Euclidean or Minkowski space to itself that leaves the ﬂat metric invariant up to a position-dependent rescaling; equivalently, the subgroup of diﬀ × Weyl that leaves invariant the ﬂat metric. In d ≥ 3 dimensions this has 12 (d + 1)(d + 2) parameters. In two dimensions it is the set of all holomorphic maps. Finite transformations require the inclusion of points at inﬁnity, as in the case of the M¨ obius transformations of the sphere. This usage has become standard in string theory and quantum ﬁeld theory, but in general relativity conformal transformation is deﬁned to be any position-dependent rescaling of the metric, now called a Weyl transformation in string theory. conifold a Calabi–Yau manifold with a singular complex structure, corresponding to the collapse of a three-cycle. The string theory on this space is singular; this is now understood to be due to the quantum eﬀects of a massless 3-brane wrapped on the cycle. conifold transition a change of topology due to condensation of massless 3brane ﬁelds. Under appropriate conditions, the potential for the massless 3-brane ﬁelds on a conifold with multiple collapsed cycles has a ﬂat direction for these ﬁelds; this corresponds to a change of topology, blowing up a 2-cycle rather than the collapsed 3-cycle. constraint a symmetry generator whose matrix elements are required to vanish in physical states, either in the BRST or OCQ sense. These can usually

Glossary

365

be understood as arising from a gauge symmetry (for so-called ﬁrst class constraints, which are all that we consider), and consist of those gauge symmetry generators that do not vanish by the equations of motion. coset CFT a CFT constructed as one of the factors of a known CFT, when the energy-momentum tensor of the latter can be written as a sum of commuting pieces. In the classic example the full CFT G is a current algebra, as is one of the factors (H). This can also be thought of as gauging the symmetry H. cosmological constant the energy density of the vacuum. In a nonsupersymmetric quantum theory (including one with spontaneously broken supersymmetry), there are many eﬀects that give rise to such an energy density. The cosmological constant problem is the problem that the cosmological constant in nature is many orders of magnitude smaller than known nonzero eﬀects. critical dimension the dimension in which a perturbative string theory is consistent in ﬂat Minkowski spacetime; the value is 26 for the bosonic string and 10 for the supersymmetric string theories. current algebra in quantum ﬁeld theory, the algebra of the currents associated with a continuous symmetry group g (or of their Fourier modes). As used here, it is the speciﬁc algebra that occurs in two-dimensional CFTs, with the energy-momentum tensor deﬁned to be of Sugawara form. The terms aﬃne Lie algebra and aﬃne Kac–Moody algebra are also used for this algebra, though like current algebra they both have broader deﬁnitions as well. The term aﬃne refers to the c-number (Schwinger) term. An untwisted current algebra is the algebra of periodic currents, with integer modes. An algebra can be twisted by any automorphism of g. cycle a topologically nontrivial submanifold (in the sense of homology); a pcycle is p-dimensional. The A- and B-cycles are a standard basis for the nontrivial one-cycles on a Riemann surface. D-brane in the type I, IIA, and IIB string theories, a dynamical object on which strings can end. The term is a contraction of Dirichlet brane. The coordinates of the attached strings satisfy Dirichlet boundary conditions in the directions normal to the brane and Neumann conditions in the directions tangent to the brane. A Dp-brane is p-dimensional, with p taking any even value in the IIA theory, any odd value in the IIB theory, and the values 1, 5, and 9 in the type I theory; a D9-brane ﬁlls space and so corresponds to an ordinary Neumann boundary condition. The Dp-brane is a source for the (p + 1)-form R–R gauge ﬁeld. The mass or tension of a D-brane is intermediate between that of an ordinary quantum or a fundamental string and that of a soliton. The low energy ﬂuctations of Dbranes are described by supersymmetric gauge theory, which is non-Abelian for coincident branes. D-instanton an object localized in (Euclidean) time as well as space, deﬁned by Dirichlet conditions on all coordinates of attached strings. This is similar to

366

Glossary a ﬁeld-theoretic instanton, corresponding to a tunneling process that changes the value of an R–R ﬁeld strength. More generally, the (p + 1)-dimensional world-volume of a Dp-brane, when localized in time and wrapped on a (p + 1)-cycle in space, has similar eﬀects.

D-string a D1-brane, in the type I and IIB string theories. D-term 1. in gauge theories with four or eight supersymmetries, the auxiliary ﬁeld in the gauge multiplet; 2 the potential term proportional to the square of this auxiliary ﬁeld, which depends only on the gauge couplings and, in the U(1) case, the value of the Fayet–Iliopoulos parameter. Del Guidice–Di Vecchia–Fubini (DDF) operators operators satisfying an oscillator algebra, which create a complete set of physical states in OCQ. descendant a state obtained by acting on a highest weight state with Virasoro raising generators. diagonal modular invariant a modular-invariant CFT formed by imposing common boundary conditions on the left- and right-moving ﬁelds. diﬀ invariance general coordinate (reparameterization) invariance, usually applied to the world-sheet coordinates. dilaton the massless scalar with gravitational-strength couplings, found in all perturbative string theories. An exactly massless dilaton would violate limits on nongravitational interactions, but a mass for the dilaton is not forbidden by any symmetry and so dynamical eﬀects will generate one in vacua with broken supersymmetry (the same holds for other moduli). The string coupling constant is determined by the value of the dilaton ﬁeld. dimensional reduction in the simplest cases, toroidal compactiﬁcation retaining only the states of zero compact momentum. More generally (and less physically) the construction of a lower-dimensional ﬁeld theory by requiring all ﬁelds to be invariant under a set of symmetries; this may have no interpretation in terms of compactiﬁcation. Dirac quantization condition for an electric and a magnetic charge, the condition that the product be quantized, µe µm = 2πn. For two dyons, which have both charges, the condition is µe µm − µm µe = 2πn. These conditions generalize to objects of dimension p and d − p − 4 in d dimensions, where one is the source of a (p + 2)-form ﬁeld strength and the other of the Poincar´e dual (d − p + 2)-form ﬁeld strength. Dirac spinor the unique irreducible representation of the algebra of Dirac matrices, which is also known as a Cliﬀord algebra. This is also a representation of the Lorentz group; in even dimensions it is reducible to two Weyl representations of the Lorentz group. The Dirac spinor is complex; in certain dimensions a further Majorana (reality) condition is compatible with Lorentz invariance. Dirichlet boundary condition the condition that the value of a ﬁeld be ﬁxed at a boundary. This is the relevant usage in string theory, but in other contexts only the tangent derivative need be ﬁxed.

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discrete torsion in forming a twisted CFT, a change in the phases of the path integral sectors and therefore in the projection on the Hilbert space. doubling trick the representation of holomorphic and antiholomorphic ﬁelds on a manifold with boundary by holomorphic ﬁelds alone, on the doubled copy of the manifold obtained by reﬂecting through the boundary. dual resonance model a phenomenological model of the strong interaction, which developed into string theory. The dual refers here to world-sheet duality. duality the equivalence of seemingly distinct physical systems. Such an equivalence often arises when a single quantum theory has distinct classical limits. One classic example is particle–wave duality, wherein a quantum ﬁeld theory has one limit described by classical ﬁeld theory and another described by classical particle mechanics. Another is the high-temperature– low-temperature duality of the Ising model. Here, low temperature is the statistical mechanical analog of the classical limit, the Boltzmann sum being dominated by the conﬁgurations of lowest energy. See Montonen–Olive duality, S-duality, string–string duality, T -duality, U-duality, world-sheet duality. eﬀective ﬁeld theory the description of a physical system below a given energy scale (or equivalently, above a given length scale). Einstein metric the metric whose leading low energy action is the Hilbert action 1 dd x (−G)1/2 R ; 2κ2 this is independent of other ﬁelds. Here κ is the gravitational coupling, related to the Planck length by κ = (8π)1/2 LP . This metric is related to other metrics such as the sigma-model metric by a ﬁeld-dependent Weyl transformation. The existence of distinct metrics would appear to violate the equivalence principle, but when the dilaton and other moduli are massive the distinction disappears. electroweak scale the mass scale of electroweak symmetry breaking, roughly 102 GeV. enhanced gauge symmetry a gauge symmetry appearing at special points in moduli space, which is not evident in the original formulation of a theory. The classic examples are the gauge symmetries that arise at special radii of toroidal compactiﬁcation, whose gauge bosons are winding states. Many other mechanisms are now known: D-branes and black p-branes wrapped on collapsing cycles, F-strings or D-strings stretched between various branes in the limit that the latter become coincident, and the gauge symmetry appearing on a zero size SO(32) instanton. Euclidean having a metric of strictly positive signature. The original connotation that the metric be ﬂat is somewhat disregarded; thus one refers to Euclidean quantum gravity, a conjectured analytic continuation of the Minkowskian theory. For the metric itself, the term Riemannian for a curved metric of Euclidean signature is more precise.

368

Glossary

Euclidean adjoint in a Euclidean quantum theory, the Hermitean adjoint combined with time-reversal. The latter operation undoes the time-reversing eﬀect of the adjoint, so that the combined operation is local. Euler number the topological invariant χ=

d

(−1)p Bp ,

p=0

where Bp is the pth Betti number. It is equal to 2(1 − g) for a Riemann surface of genus g. More properly, the Euler characteristic. expectation value a path integral with speciﬁed insertions. This is the term that we have chosen to use, but correlation function and correlator are also in common usage. extended chiral algebra the full set of holomorphic operators in a CFT. extended supersymmetry a supersymmetry algebra in which the supercharges comprise more than one copy of the smallest spinor representation for the given spacetime dimension. F -term 1. the auxiliary ﬁeld in the chiral multiplet of d = 4, N = 1 supersymmetry; 2. the potential term proportional to the square of this ﬁeld. F theory 1. a systematic description of IIB superstring states with nontrivial dilaton and R–R scalar backgrounds, which relates these ﬁelds to the modulus τ of an auxiliary two-torus; 2. a conjectured twelve-dimensional quantum theory underlying the IIB string. The name is inspired by Mtheory, with F for father. Faddeev–Popov determinant the Jacobian determinant arising from the reduction of a gauge-invariant functional integral to an integral over a gauge slice. Faddeev–Popov ghosts the wrong-statistics quantum ﬁelds used to give a functional integral representation of the Faddeev–Popov determinant. Fayet–Iliopoulos term in U(1) gauge theories with four or eight supersymmetries, a term in the action which is linear in the auxiliary D-term ﬁeld. Feigin–Fuchs representation a representation of minimal model expectation values in terms of free ﬁelds. ﬁbration a space which is locally the product of a ﬁber F and a base B. The geometry of the ﬁber varies as one moves over the base, and may become singular. Typical ﬁbers are tori and the K3 manifold. ﬁrst Chern class on a complex manifold, the Dolbeault cohomology class of the Ricci form Ri¯ dz i d¯z¯ . ﬁrst-quantized description the representation of a quantized particle theory as a sum over particle paths, or of a string theory as a sum over worldsheets. Second-quantized refers to the representation in terms of a functional integral over ordinary or string ﬁelds. The term second-quantized implies

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369

the reinterpretation of the ﬁrst-quantized wavefunction as a ﬁeld operator. This terminology is in common usage, but it has been argued that it is unsatisfactory, in that it implies a deep principle where none may exist. Since the sum over world-sheets is itself a quantum ﬁeld theory, one can equally well call it second-quantized, in which case string ﬁeld theory is thirdquantized. Third quantization of an ordinary ﬁeld theory would describe operators that create and destroy universes, a concept which may or may not be useful. Fischler–Susskind mechanism the cancellation of divergences and anomalies in the world-sheet quantum ﬁeld theory against divergences and anomalies from integration over small topological features at higher orders of string perturbation theory. This is needed for the consistency of string perturbation theory in a quantum-corrected background. ﬁxed point 1. (in geometry) a point left invariant by a given symmetry transformation. This becomes a boundary point or a singularity if the space is identiﬁed under the transformation; 2. (in quantum ﬁeld theory) a quantum theory whose physics is independent of length scale (scale-invariant). Usually such a theory is conformally invariant as well. A UV ﬁxed point is the theory governing the short-distance physics of a quantum ﬁeld theory; an IR ﬁxed point is the theory governing the long-distance physics of a quantum ﬁeld theory. A trivial IR ﬁxed point has no massless ﬁelds. A nontrivial IR ﬁxed point has massless ﬁelds with nonvanishing interactions. A theory whose IR limit is a massless free ﬁeld theory is therefore described by neither of these terms; it is a noninteracting IR ﬁxed point. ﬂat direction in scalar ﬁeld space, a line of degenerate local minima. The ﬁeld corresponding to this direction is a modulus. ﬂop a change of topology which can occur in weakly coupled string theory, where a two-cycle collapses and then a diﬀerent two-cycle blows up. fractional charge an unconﬁned particle whose electric charge is not a multiple of that of the electron. These exist in most d = 4 string theories, although in many cases all are superheavy. fractional string theory a proposed generalization of string theory having constraints whose spin is not a multiple of 12 . No complete construction exists. Fuchsian group a discrete subgroup Γ of the SL(2, R) M¨ obius transformations of the complex upper-half-plane H, with additional conditions such that H/Γ is a manifold and in particular a Riemann surface. functional integral in our usage, synonymous with path integral. fundamental region in relation to a coset space M/Γ where Γ is a discrete group, a region F such that every point in M is identiﬁed with exactly one point in the interior of F or with one or more points on the boundary of F. fundamental string (F-string) the original string whose quantization deﬁnes a weakly coupled string theory, as distinguished from D-strings and solitonic strings.

370

Glossary

fusion rule the speciﬁcation of which conformal families appear in the operator product of any two primary ﬁelds in a given CFT. gauge-ﬁxing the reduction of a redundant (gauge-invariant) description of a quantum theory to a description with a single representative from each equivalence class. gaugino a spin- 12 fermion in the same supersymmetry multiplet as a gauge boson. gaugino condensation a strong coupling eﬀect where a product of gaugino ﬁelds acquires a vacuum expectation value. This generally breaks a chiral symmetry but does not directly break supersymmetry; however, in combination with other ﬁelds it often induces supersymmetry breaking. generation a family of quarks and leptons, described by spinor ﬁelds in a chiral but anomaly-free set of SU(3) × SU(2) × U(1) representations. In SU(5) grand uniﬁcation these become a 5 + 10, in SO(10) they are contained in a 16, and in E6 they are contained in a 27. An antigeneration is the conjugate representation; the distinction between generation and antigeneration is a matter of convention. genus the number g of handles on a closed oriented Riemann surface: g = 0 is a sphere, g = 1 is a torus, and so on. Gepner model a string model based on N = 2 minimal model CFTs. ghosts see Faddeev–Popov ghosts. Gliozzi–Scherk–Olive (GSO) projection a construction of modular-invariant string theories by summing over R and NS boundary conditions on the fermion ﬁelds and projecting onto states of deﬁnite world-sheet fermion number. In supersymmetric string theories there are independent GSO projections on the left-movers and right-movers. The diagonal projection, which acts simultaneously on both sides, produces a nonsupersymmetric theory. goldstino the massless spin- 12 Goldstone fermion associated with spontaneously broken supersymmetry. In supergravity it combines with the gravitino to form a massive fermion. Goldstone boson the massless scalar corresponding to ﬂuctuations of the direction of spontaneous symmetry breaking. grand uniﬁcation the uniﬁcation of the SU(3)×SU(2)×U(1) gauge symmetries in a simple group. grand uniﬁcation scale the mass scale of spontaneous breaking of the grand uniﬁed group. Proton stability and the uniﬁcation of the couplings require that it be within two or three orders of magnitude of the gravitational scale. Grassmann variable the elements θi of an algebra with the relation θi θj = −θj θi . These are used to give a path integral representation of fermionic ﬁelds, and to deﬁne superspace. They are also called anticommuting cnumbers.

Glossary

371

gravitational scale the mass scale at which the dimensionless gravitational coupling becomes of order 1, mgrav = κ−1 = 2.4 × 1018 GeV; this is (8π)−1/2 times the Planck mass. gravitino a spin- 23 fermion in the same supersymmetry multiplet as the graviton. Green–Schwarz mechanism the cancellation of an anomaly by the modiﬁed transformation law of a p-form potential in a Chern–Simons term. Green–Schwarz superstring a manifestly supersymmetric formulation of the supersymmetric string theories, with a spacetime-fermionic gauge invariance known as κ symmetry. There is no simple covariant gauge ﬁxing. H-monopole a monopole carrying the magnetic charge of the antisymmetric tensor gauge ﬁeld Bµn . heterotic 5-brane the 5-brane carrying the magnetic charge of the massless heterotic string 2-form potential. It is obtained as the limit of a zero size instanton in the heterotic string gauge ﬁelds. The instanton conﬁguration is localized in four spatial dimensions, and is therefore a 5-brane in nine spatial dimensions. heterotic string a string with diﬀerent constraint algebras acting on the leftand right-moving ﬁelds. The case of phenomenological interest has a (0, 1) superconformal constraint algebra, with spacetime supersymmetry acting only on the right-movers and with gauge group E8 × E8 or SO(32). Hagedorn temperature the temperature at which the thermal partition function of free strings diverges, due to the exponential growth of the density of states of highly excited strings. hidden sector the ﬁelds that couple to the Standard Model only through gravitational-strength interactions. In hidden sector models, these include the ﬁelds responsible for supersymmetry breaking. highest weight state in CFT, a state annihilated by all Virasoro lowering operators, or more generally by all lowering operators in a given algebra. Hodge number the number of nontrivial (p, q)-forms in Dolbeault cohomology, denoted hp,q . holographic principle the conjecture that the states of quantum gravity in d dimensions have a natural description in terms of a (d − 1)-dimensional theory. This radical departure from local ﬁeld theory was motivated by the black hole information problem, and has played a role in attempts to formulate M-theory. holomorphic analytic, as used in the theory of complex variables. The Minkowskian continuation is left-moving. An antiholomorphic ﬁeld is analytic in the conjugate variable, and its continuation is right-moving. holomorphic quadratic diﬀerential a globally deﬁned holomorphic (2, 0)-form on a Riemann surface.

372

Glossary

holomorphic vector ﬁeld a globally deﬁned holomorphic (−1, 0)-form on a Riemann surface. holonomy consider the parallel transport of a vector around a closed loop on a d-dimensional manifold: it returns to an O(n) rotation of its original value. The set of all rotations that are obtained in this way for a given manifold is a subgroup of O(n); this is the holonomy group. homology see cohomology. hyperelliptic surface a Riemann surface with a Z2 symmetry. This can be represented as a two-sheeted cover of the sphere with branch cuts. hyper-K¨ahler manifold a 4k-dimensional manifold of holonomy Sp(k) ⊂ SO(4k). This is the geometry of the moduli space of hypermultiplets in d = 6, N = 1 or d = 4, N = 2 supersymmetry, in the limit in which gravity decouples. hypermultiplet in d = 6, N = 1 or d = 4, N = 2 supersymmetry, the multiplet whose bosonic content is four real massless scalars. identify to deﬁne two points (or other objects) to be equivalent, thus producing a coset space. infrared (IR) divergence a divergence arising from long distances in spacetime, usually signifying that one has calculated the wrong thing. inheritance principle in twisted (orbifold) theories, the principle that the treelevel amplitudes of untwisted states are the same as in the untwisted theory. insertion the integrand of a path integral, excluding the weight exp(iS) or exp(−S). instanton in a Euclidean path integral, a nonconstant conﬁguration that is a local but not a global minimum of the action. Such conﬁgurations are usually localized in spacetime, are usually topologically nontrivial, and are of interest when they give rise to eﬀects such as tunneling that are not obtained from small ﬂuctuations around a constant conﬁguration. Spacetime instantons are instantons in the eﬀective ﬁeld theory in spacetime. Worldsheet instantons are instantons in the world-sheet quantum ﬁeld theory, and correspond to world-sheets wrapping around nontrivial two-cycles of spacetime. intersection number the number of points at which a set of surfaces intersect, weighted by the orientation of the intersection. irrelevant interaction an interaction whose dimensionless strength decreases with increasing length scale. In perturbation theory, this is equivalent to a nonrenormalizable interaction. K3 manifold the unique nontrivial Calabi–Yau manifold of four (real) dimensions. To be precise, it is topologically unique, but possesses complex structure and K¨ ahler moduli. Its holonomy is SU(2), so that half of the supersymmetries of a theory are broken upon compactiﬁcation on K3.

Glossary

373

Kac determinant the determinant of the matrix of inner products of states at a given L0 level of a Verma module. Kac–Moody algebra see current algebra. K¨ahler form the (1, 1)-form Gi¯ dz i d¯z¯ , formed from a K¨ahler metric on a complex manifold. K¨ahler manifold a complex manifold of U(n) holonomy in n complex dimensions. K¨ahler moduli the moduli parameterizing the K¨ahler form. K¨ahler potential the potential K(z, ¯z ), in terms of which the metric of a K¨ahler manifold is determined, Gi¯ = ∂i ∂¯ K. This is not globally deﬁned, being determined only up to a K¨ahler transformation K(z, ¯z ) → K(z, ¯z ) + f(z) + f(z)∗ . Kaluza–Klein gauge ﬁeld in a compactiﬁed theory, a gauge ﬁeld originating from the metric of the higher-dimensional theory. The gauge group is the isometry group of the compact space. Kaluza–Klein monopole a monopole carrying the magnetic charge of a U(1) Kaluza–Klein gauge symmetry. The monopole conﬁguration is the smooth Taub–NUT spacetime. It is localized in three spatial dimensions, and is therefore a 6-brane in nine spatial dimensions. Kaluza–Klein states states with nonzero momentum in a compact spatial direction. Knizhnik–Zamolodchikov (KZ) equation the diﬀerential equation determining the expectation values of the primary ﬁelds of a current algebra. Landau–Ginzburg model a scalar ﬁeld theory which has a nontrivial IR ﬁxed point when the potential is appropriately tuned. In particular, this gives a Lagrangian representation of the minimal model CFTs. large coordinate transformation a coordinate transformation that is not continuously connected to the identity. lattice the set Γ of integer linear combinations of n linearly independent basis vectors in n dimensions. Given a Euclidean or Lorentzian metric, an even lattice is one whose points have even length-squared. The dual lattice Γ∗ is the set of points v such that v · w ∈ Z for all w ∈ Γ. The root lattice is the set of integer linear combinations of the roots of a Lie algebra. level 1. the quantized c-number term in a current algebra, also known as the Schwinger term; 2. the total oscillator excitation number in free ﬁeld theory; 3. in a conformal family, the diﬀerence between the L0 eigenvalue of a given state and that of the highest weight state. ˜ 0 ∈ Z. level-matching the modular invariance condition that L0 − L Lie algebra an algebra with an antisymmetric product that satisﬁes the Jacobi identity. A simple Lie algebra has no subalgebra that commutes with its complement. A simply-laced Lie algebra has all roots of equal length. A

374

Glossary graded Lie algebra has odd and even elements, with a symmetric product between odd elements.

light-cone gauge in string theory, the choice of world-sheet time coordinate to coincide with a particular spacetime null coordinate. In theories with local symmetries, a gauge choice such that the connection in a given null direction vanishes. linear dilaton theory a scalar CFT in which the energy-momentum tensor includes a term proportional to the second derivative of the scalar. This arises in string theory when the dilaton is a linear function of position. linear sigma model a scalar ﬁeld theory whose kinetic term is ﬁeld independent, but whose long-distance physics is governed by a nonlinear sigma model. Liouville ﬁeld theory the CFT of a scalar ﬁeld with an exponential interaction. This arises in various situations, including the noncritical string. It corresponds to bosonic string theory in a linear dilaton plus exponential tachyon background. little string theory one of several interacting string theories without gravity, notably found on NS5-branes in the limit of zero string coupling. loop expansion in quantum ﬁeld theory, the Feynman graph expansion, which is equivalent to the expansion in powers of ¯h. The string loop expansion is the sum over Riemann surfaces, with dimensionless string coupling g. The world-sheet loop expansion is the nonlinear sigma model perturbation expansion, in powers of α /Rc2 with Rc the compactiﬁcation radius. Lorentzian having a mixed signature (−, . . . , −, +, . . . , +). lowering operator operators that reduce the energy of a given state. In CFT, ˜ 0 ) by n units carry a operators that reduce the Virasoro generator L0 (or L grading (subscript) n. M-theory 1. (narrow) the limit of strongly coupled IIA theory with elevendimensional Poincar´e invariance; 2. (broad; most common usage) the entire quantum theory whose limits include the various weakly coupled string theories as well as M-theory in the narrow sense. The name is deliberately ambiguous, reﬂecting the unknown nature of the theory; M has variously been suggested to stand for membrane, matrix, mother, and mystery. M2-brane the 2-brane of M-theory, which couples to the potential A3 of eleven-dimensional supergravity. M5-brane the 5-brane of M-theory, which carries the magnetic charge of the potential A3 of eleven-dimensional supergravity. macroscopic string a string whose length is much greater than the characteristic string length scale. In particular, it is sometimes useful to consider an inﬁnite string stretching across spacetime. Majorana condition a Lorentz-invariant reality condition on a spinor ﬁeld. This can be imposed only if the spacetime dimension is 1, 2, 3, 4, or 8 (mod 8).

Glossary

375

marginal interaction an interaction whose dimensionless strength is independent of the length scale. In general this might hold only to ﬁrst order in the coupling of the interaction; a truly marginal interaction is one that remains marginal even with ﬁnite coupling. matrix models quantum mechanical systems with matrix degrees of freedom, with critical points governed by noncritical string theories. matrix theory a quantum mechanical system with matrix degrees of freedom and 32 supercharges, obtained by dimensional reduction of d = 10 supersymmetric U(n) Yang–Mills theory. In the large-n limit this is conjectured to deﬁne M-theory (in the broad sense). minimal models several families of solvable CFTs, in which every conformal family is degenerate. There are inﬁnite series of unitary minimal models having N = 0, N = 1, and N = 2 superconformal symmetries, which converge from below on the central charges 1, 32 , and 3 respectively. Minkowskian having a signature (−, +, +, . . . , +). mirror symmetry an equivalence between string theories compactiﬁed on distinct manifolds. The equivalence reverses the sign of the U(1) charge of one N = 2 superconformal algebra, and therefore changes the sign of the Euler number of the manifold. M¨obius group the globally deﬁned SL(2, C) conformal symmetry of the sphere; or, the globally deﬁned SL(2, R) conformal symmetry of the disk. mode operators the spatial Fourier components of a quantum ﬁeld. model see vacuum. modular group the group of large coordinate transformations (often applied to a Riemann surface but also applicable to spacetime). modular invariance the invariance of the string path integral under large coordinate transformations. moduli 1. the parameters labeling the geometry of a manifold. Notable examples are the parameters for the complex structure of the string world-sheet, and the parameters for the geometry of compactiﬁcation; 2. the parameters labeling a space of degenerate (and, usually, physically inequivalent) vacua in quantum ﬁeld theory. This is closely related to the compactiﬁcation example: in expanding around the classical limit, each compact solution of the ﬁeld equations gives a vacuum of the quantum theory, to leading order; 3. the massless ﬁelds corresponding to position dependence of these parameters. Contrast Goldstone boson. moduli space the space of geometries or vacua, whose coordinates are the moduli. monodromy for a quantity which is locally single-valued, the multi-valuedness around nontrivial closed paths. Montonen–Olive duality the weak–strong duality of d = 4, N = 4 Yang–Mills theory.

376

Glossary

Nambu–Goto action a string action, which is proportional to the invariant area of the world-sheet in spacetime. Narain compactiﬁcation the abstract description of toroidal compactiﬁcation in terms of the lattice of left- and right-moving momenta. naturalness problem the problem of explaining why a constant of nature takes a value much smaller than estimated nonzero contributions. Examples are the Higgs scalar mass, the cosmological constant, and the QCD θ-angle. Neumann boundary condition the condition that the normal derivative of a ﬁeld vanish at a boundary; the value of the ﬁeld is free to ﬂuctuate. Neveu–Schwarz algebra the world-sheet algebra of the Fourier modes of the supercurrent and energy-momentum tensor, in a sector where the supercurrent is antiperiodic and its moding therefore half-integer-valued. Neveu–Schwarz (NS) boundary condition the condition that a fermionic ﬁeld on the world-sheet be antiperiodic, in the closed string or in the double of the open string (see doubling trick). Its Fourier moding is then half-integervalued. Neveu–Schwarz 5-brane in the type I and type II superstring theories, the 5-brane that carries the magnetic charge of the NS–NS 2-form potential. Neveu–Schwarz–Neveu–Schwarz (NS–NS) states in type II super-string theories, the bosonic closed string states whose left- and right-moving parts are bosonic. These include the graviton and dilaton, and in the type II case a 2-form potential. no-ghost theorem 1. the theorem that the OCQ or BRST Hilbert space has a positive inner product; 2. the further theorem that the string amplitudes are well deﬁned and unitary in this space. no-scale model a ﬁeld theory that has, to some approximation, a line of degenerate vacua with broken supersymmetry. Noether’s theorem the theorem that an invariance of the Lagrangian implies a conserved quantity. noncommutative geometry a generalization of ordinary geometry, focusing on the algebra of functions on a space. The noncommutative collective coordinates of D-branes suggest the need for such a generalization. noncritical string theory 1. a Weyl-noninvariant string theory — one with a measure of world-sheet distance that is independent of the embedding in spacetime. These include strings with an independent world-sheet metric ﬁeld, and strings with a short-distance cutoﬀ; 2. more recently, the term has been applied to any string theory that does not have a weakly coupled limit with a Weyl-invariant world-sheet theory. In this form it includes various theories with stringlike excitations in which the coupling is ﬁxed to be of order 1, such as the (2,0) theory and the little string theories. Such a theory does not have a well-deﬁned world-sheet, because processes that change the world-sheet topology cannot be turned oﬀ.

Glossary

377

nonlinear sigma model a scalar ﬁeld theory in which the kinetic term has a ﬁeld-dependent coeﬃcient. This has a natural interpretation in terms of a curved ﬁeld space, and corresponds to string theory in curved spacetime, or more generally one with position-dependent background ﬁelds. nonrenormalization theorem a theorem restricting the form of quantum corrections to a given amplitude, or to the eﬀective action. It may require that these corrections vanish, that they arise only at speciﬁc orders of perturbation theory, or that they arise only nonperturbatively. normal ordering a prescription for deﬁning products of free ﬁelds by speciﬁc subtractions of divergent terms. Conformal normal ordering, denoted : :, produces operators with simple conformal properties. Creation–annihilation normal ordering, denoted ◦◦ ◦◦ , where lowering operators are put to the right of raising operators, produces operators with simple matrix elements. Boundary normal ordering, denoted , is conformal normal ordering with an additional image charge subtraction to produce operators that are ﬁnite as they approach a boundary. null state a physical state that is orthogonal to all physical states including itself. Or, a descendant in a conformal family which is orthogonal to all states in the family. old covariant quantization (OCQ) a method of quantizing string theory, similar to the Gupta–Bleuler quantization of electrodynamics. It is equivalent to the light-cone and BRST quantizations. one-loop the leading quantum correction, coming from surfaces of Euler number zero in string perturbation theory. open string a string that is topologically a line segment. operator equation in quantum theory, an equality between operators that holds in arbitrary matrix elements; equivalently, an equality that holds when inserted into a functional integral with arbitrary boundary conditions. operator product expansion (OPE) the expansion of a product of operators as a sum of local operators. This provides an asymptotic expansion, as the separation of the operators vanishes, for an arbitrary expectation value containing the product. In CFT the expansion is convergent. orbifold 1. (noun) a coset space M/H, where H is a group of discrete symmetries of a manifold M. The coset is singular at the ﬁxed points of H; 2. (noun) the CFT or string theory produced by the gauging of a discrete world-sheet symmetry group H. If the elements of H are spacetime symmetries, the result is a theory of strings propagating on the coset space M/H. A nonAbelian orbifold is one whose point group is non-Abelian. An asymmetric orbifold is one where H does not have a spacetime interpretation and which in general acts diﬀerently on the right-movers and left-movers of the string; 3. (verb) to produce such a CFT or string theory by gauging H; this is synonymous with the third deﬁnition of twist.

378

Glossary

oriented string theory a string theory in which the world-sheet has a deﬁnite orientation; world-sheet parity-reversal is not treated as a gauge symmetry. orientifold a string theory produced by the gauging of a world-sheet symmetry group H, where H includes elements that combine the world-sheet parityreversal Ω with other symmetries. orientifold plane a plane (of any dimension p) consisting of ﬁxed points of the orientifold group H (speciﬁcally, of an element of H that includes Ω). p-brane a p-dimensional spatially extended object. Examples are black p-branes, Dp-branes, M2- and M5-branes, NS5-branes, heterotic 5-branes, and (in d > 4) Kaluza–Klein monopoles. p-form a fully antisymmetric p-index tensor, usually written in an index-free notation. p-form gauge ﬁeld a generalization of Abelian gauge theory, with a p-form potential A, a (p − 1)-form gauge parameter λ, and a (p + 1)-form ﬁeld strength F. For p = 0 this is an ordinary massless scalar; for p = 1 it is an Abelian gauge ﬁeld. A (p + 1)-form potential couples naturally to a p-brane, through the integral of the form over the world-volume. A self-dual p-form theory is one where ∗F = F; this requires the spacetime dimension to be d = 2p + 2, and further d must be 2 mod 4 (in the Minkowskian case) in order that ∗∗ = 1. p-p string an open string with one endpoint on a Dp-brane and the other on a Dp -brane. (p, q)-form on a complex manifold, a tensor that is completely antisymmetric in p holomorphic indices and q antiholomorphic indices. (p, q) string a bound state of p F-strings and q D-strings in the IIB theory. parafermion CFT a family of coset CFTs with Zn symmetry, generalizing the Z2 -invariant free fermion theory. These describe the generic critical behavior of a system with Zn symmetry. parity transformation an operation that reﬂects one spatial dimension, or any odd number. One distinguishes spacetime parity, P (or β) and world-sheet parity Ω. partition function a sum over the spectrum of a quantum system, weighted by e−H/T where H is the Hamiltonian and T the temperature. Often additional charges are included in the exponent. This is the basic object in equilibrium statistical mechanics. In string theory it is given by a path integral on the torus or the cylinder, and so arises in one-loop amplitudes. path integral a representation of the transition amplitudes of a quantum system as a coherent sum over all possible histories. In quantum mechanics the history is a particle path; in quantum ﬁeld theory it is a path in ﬁeld space; in ﬁrst-quantized string theory it is the embedding of the string world-sheet in spacetime.

Glossary

379

Pauli–Villars regulator a means of regulating quantum ﬁeld theories by introducing a very massive wrong-statistics ﬁeld. Peccei–Quinn (PQ) symmetry an approximate symmetry, violated only by anomalies. period matrix a g × g matrix characterizing the complex structure of a genus-g Riemann surface. perturbation theory the expansion of the amplitudes of a quantum system in powers of the coupling. physical state in a quantum system with constraints, a state annihilated by the constraints. In OCQ this is a state annihilated by the Virasoro lowering generators and having a speciﬁed L0 eigenvalue.. In BRST quantization it is a state annihilated by the BRST operator. In both these cases, the true physical spectrum is the space of physical states with an additional equivalence relation, physical states diﬀering by a null state being identiﬁed. picture in the RNS superstring, one of several isomorphic representations of the vertex operators. The q-picture consists of vertex operators of βγ ghost charge q. The natural pictures are q = −1 and − 12 , with higher pictures including partial integrations over supermoduli space. The picture changing operator increases q by one. Planck length the natural length scale of quantum gravity, LP = MP−1 = 1.6 × 10−33 cm, constructed from ¯h, c, and GN . Planck mass the natural mass scale of quantum gravity, MP = 1.22 × 1019 GeV, constructed from ¯h, c, and GN . plumbing ﬁxture a procedure for constructing higher genus Riemann surfaces from lower ones by sewing in a handle. The construction includes a parameter q, such that when q goes to 0 the handle degenerates or pinches. This gives a canonical representation of the boundary of the moduli space of Riemann surfaces. Poincar´e dual a map from p-forms to (d − p)-forms, given by contraction with the completely antisymmetric tensor. Poincar´e invariance the invariance group of the ﬂat metric, consisting of translations and Lorentz transformations. point group the orbifold group H, with translations ignored (applicable only for orbifolds having a spacetime interpretation). Polyakov path integral a representation of ﬁrst-quantized string theory as a path integral with an independent world-sheet metric. A local Weyl symmetry guarantees that the classical degrees of freedom are the same as those of the Nambu–Goto theory. primary ﬁeld in CFT, a local operator annihilated by all of the lowering generators of a given algebra, such as the Virasoro algebra. The corresponding state is a highest weight state.

380

Glossary

projective space a compact n-dimensional space constructed from a linear (n + 1)-dimensional space by identifying points under the overall rescaling (x1 , . . . , xn+1 ) ∼ = (λx1 , . . . , λxn+1 ). For xi and λ real this produces RP n , and for xi and λ complex it produces CP n (which has n complex = 2n real dimensions). pseudospin in a current algebra, an SU(2) subalgebra not contained in the Lie algebra of the center-of-mass modes. puncture a marked point on a Riemann surface, the position of a vertex operator. QCD string a reformulation of non-Abelian gauge theory as a string theory, conjectured to exist at least in the limit of a large number of colors. quaternionic manifold a 4k-dimensional manifold with the holonomy group Sp(k) × SU(2) ⊂ SO(4k), with speciﬁc SU(2) curvature. This is the geometry of the moduli space of hypermultiplets in d = 6, N = 1 or d = 4, N = 2 supergravity. R symmetry a symmetry that acts nontrivially on the supercurrent(s). raising operator operators that reduce the energy of a given state. In CFT, ˜ 0 ) by n units carry operators that increase the Virasoro generator L0 (or L a grading (subscript) −n. Ramond algebra the world-sheet algebra of the Fourier modes of the supercurrent and energy-momentum tensor, in a sector where the supercurrent is periodic and its moding therefore integer-valued. Ramond (R) boundary condition the condition that a fermionic ﬁeld on the world-sheet be periodic, in the closed string or in the double of the open string (see doubling trick). Its Fourier moding is then integer-valued. Ramond–Neveu–Schwarz (RNS) superstring the formulation of type I and II superstrings that has superconformal invariance but not manifest spacetime supersymmetry. The latter emerges after imposing the GSO projection on the string Hilbert space. Ramond–Ramond (R–R) states in type I and type II superstring theories, the bosonic closed string states whose left- and right-moving parts are fermionic. These include p-form potentials Cp , with p taking all odd values in the IIA string and all even values in the IIB string. rank 1. the maximal number of commuting generators of a Lie algebra; 2. the number of indices on a tensor. rational CFT a CFT with a ﬁnite number of primary ﬁelds under an extended chiral algebra, a generalization of the minimal models. Such CFTs are highly constrained. refermionization after bosonization, the construction of new spin- 12 ﬁelds from linear combinations of the bosonic ﬁelds. These fermions are nonlocal, nonlinear functions of the original fermions.

Glossary

381

Regge behavior in scattering at large center-of-mass energy-squared s and ﬁxed momentum transfer-squared −t, the scaling of the amplitude as sα(t) . The values of t where α(t) = j is a nonnegative integer correspond to exchange of a particle of spin j and mass-squared −t. Regge slope denoted α , the square of the characteristic length scale of perturbative string theory. The tension of the fundamental string is 1/2πα . relevant interaction an interaction whose dimensionless strength increases with distance. In perturbation theory, this is equivalent to a superrenormalizable interaction. renormalization group equation the diﬀerential equation governing the change of physics with length scale. renormalization theory the calculus of path integrals. Riemann–Roch theorem the theorem that the number of metric moduli minus the number of conformal Killing vectors on a Riemann surface is −3χ, with χ being the Euler number of the surface; and, generalizations of this result. Riemann surface a two-(real)-dimensional complex manifold, equivalent to a Weyl equivalence class of Riemannian manifolds. root a vector of the eigenvalues of the maximal set of commuting generators of a Lie algebra, associated with a state in the adjoint representation. S-duality a duality under which the coupling constant of a quantum theory changes nontrivially, including the case of weak–strong duality. Important examples are the SL(2, Z) self-dualities of IIB string theory and of d = 4, N = 4 supersymmetric Yang–Mills theory. More loosely, it is used for weak–strong dualities between diﬀerent theories, such as IIA–M-theory (on a circle) duality, SO(32) heterotic–type I duality, and E8 heterotic–M-theory (on an interval) duality. In compactiﬁed theories, the term S-duality is limited to those dualities that leave the radii invariant, up to an overall coupling-dependent rescaling; contrast T-duality and U-duality. S-matrix the overlap amplitude between states in the inﬁnite past and states in the inﬁnite future; the scattering amplitude. In coordinate-invariant quantum theories this is generally the simplest invariant. The term usually implies a basis of free particle states; this is problematic in theories with massless particles due to IR divergences, and meaningless in theories at nontrivial IR ﬁxed points. scale transformation a rigid rescaling of spacetime, or of the world-sheet. Scherk–Schwarz mechanism the breaking of supersymmetry by dimensional reduction that includes a spacetime rotation. Schottky group a discrete subgroup Γ of the SL(2, C) M¨ obius transformations of the sphere S2 , with additional conditions such that S2 /Γ is a manifold and in particular a Riemann surface. Schwarzian the combination of derivatives appearing in the ﬁnite conformal transformation of the energy-momentum tensor.

382

Glossary

Schwinger–Dyson equation the operator equations of a quantum theory, expressed as equations for the expectation values. short multiplet see BPS state. sigma model metric the metric appearing in the string world-sheet action. Also known as the string metric, this diﬀers from the Einstein metric by a dilatondependent Weyl transformation. simple current an operator J such that, for any primary ﬁeld O, the operator product JO contains only a single conformal family. soliton a state whose classical limit is a smooth, localized, and (usually) topologically nontrivial classical ﬁeld conﬁguration; this includes particle states, which are localized in all directions, as well as extended objects. By contrast, a state of ordinary quanta is represented near the classical limit by small ﬂuctuations around a constant conﬁguration. In a theory with multiple classical limits (dualities), solitons and quanta may exchange roles. special K¨ahler geometry the geometry of the moduli space of vector multiplets in d = 4, N = 2 supergravity. It is most simply deﬁned (section B.7) in terms of special coordinates, which are ﬁxed up to a symplectic transformation. Rigid special geometry is obtained in the limit where gravity decouples and the supersymmetry becomes global. spectral ﬂow the adiabatic change in the spectrum produced by a continuous change in the boundary conditions. spin the behavior of a ﬁeld or a state under rotations. In a CFT this is given in terms of the conformal weights by h − ˜h. spin ﬁeld the vertex operator for a Ramond ground state, which produces a branch cut in the spinor ﬁelds. spin structure one of a set of inequivalent ways of deﬁning a spinor ﬁeld globally on a manifold. Roughly speaking, it corresponds to a choice of signs in the square roots of the transition functions. spurious state in OCQ, a state produced by Virasoro raising operators. state–operator isomorphism in CFT, a one-to-one isomorphism between states of the theory quantized on a circle and local operators. Also, a one-toone isomorphism between states of the theory quantized on an interval and local operators on a boundary. In d dimensions the circle becomes a (d − 1)-sphere and the interval a (d − 1)-hemisphere. string coupling the dimensionless parameter g governing the weights of diﬀerent Riemann surfaces in string perturbation theory, the contribution from surfaces of Euler number χ being weighted by g −χ . The string coupling is related to the dilaton by g = eΦ . This deﬁnition corresponds to the amplitude to emit a closed string; the amplitude to emit an open string is proportional to g 1/2 . string ﬁeld theory the representation of string theory as theory of ﬁelds, the ﬁelds being maps from a circle (or interval) into spacetime. This corresponds

Glossary

383

to an inﬁnite number of ordinary quantum ﬁelds. This formalism can reproduce string perturbation theory, but it is unclear whether it can be deﬁned beyond perturbation theory. string metric see sigma model metric. string scale the mass scale α−1/2 characterizing the tower of string excitations. string–string duality a term sometimes used to denote a weak–strong duality between diﬀerent string theories, in particular between the heterotic string compactiﬁed on T 4 and the IIA string compactiﬁed on K3. string tension the mass per unit length of a string at rest, related to the Regge slope by 1/2πα . Sugawara construction in current algebra, the construction of the energymomentum tensor as a product of two currents. Originally proposed as a phenomenological model in four dimensions, this was later found to be an exact result in two dimensions. superconformal algebra an extension of the conformal (Virasoro) algebra to in˜ superconformal algebra clude anticommuting spinor generators. The (N, N) ˜ has N left-moving and N right-moving supercurrents. superconformal current algebra an extension of the conformal transformations to include both spin- 23 and spin-1 currents. superconformal ﬁeld theory (SCFT) a quantum ﬁeld theory that is invariant under superconformal transformations. supercurrent a conserved spinor current. This includes the world-sheet current TF associated with superconformal transformations, and the spacetime current associated with spacetime supersymmetry. superﬁeld a ﬁeld on superspace, with speciﬁc transformation properties under a change of coordinates. supergravity the union of general relativity and supersymmetry, implying also the promotion of supersymmetry to a local symmetry. supermanifold (or superspace) a formal extension of the concept of manifold to include both commuting and anticommuting (Grassmann) coordinates. supermoduli the anticommuting parameters characterizing a super-Riemann surface. superpartner scale the mass scale of the superpartners of the Standard Model particles. This is expected to be between 102 and 103 GeV if supersymmetry solves the naturalness problem of the Higgs scalar mass. superpotential in d = 4, N = 1 supersymmetry, the holomorphic function of the superﬁelds that determines the nongauge interactions. super-Riemann surface a supermanifold deﬁned in terms of superconformal transition functions between patches. supersymmetry a symmetry whose charge transforms as a spinor, which relates the masses and couplings of fermions and bosons.

384

Glossary

supersymmetry breaking scale the mass scale of the expectation value that breaks supersymmetry. The superpartner scale is the supersymmetry breaking scale times the strength of the coupling of the Standard Model ﬁelds to the supersymmetry breaking ﬁelds. T -duality a duality in string theory, usually in a toroidally compactiﬁed theory, that leaves the coupling constant invariant up to a radius-dependent rescaling and therefore holds at each order of string perturbation theory. Most notable is R → α /R duality, which relates string theories compactiﬁed on large and small tori by interchanging winding and Kaluza–Klein states. More generally it includes shifts of antisymmetric tensor backgrounds and large coordinate transformations in spacetime. Contrast S-duality and U-duality. ’t Hooft–Polyakov monopole a classical solution with magnetic charge, which exists whenever a simple group is spontaneously broken to a group with a U(1) factor. tachyon a particle (almost always a scalar) with a negative mass-squared, signifying an instability of the vacuum. tadpole an amplitude for creation of a single particle from the vacuum, induced by quantum eﬀects. target space the space in which a function takes its values. This is usually applied to the nonlinear sigma model on the string world-sheet, where the target space is itself spacetime. Teichm¨uller parameters the moduli for the complex structure of a Riemann surface (strictly speaking, points in Teichm¨uller space are not identiﬁed under the modular group). tensionless string theory an interacting theory with tensionless strings. These can arise as p-branes with p − 1 directions wrapped on a collapsing cycle, as various 2-branes with one direction stretched between higher-dimensional branes when the latter become coincident, and on zero-size E8 × E8 instantons. In general the coupling is ﬁxed to be of order one, so there is no perturbation expansion. Tensionless implies that the string tension in units of the gravitational scale goes to zero; it is not applied to the fundamental string, which becomes noninteracting in that limit. tensor multiplet 1. the multiplet of d = 6, (1, 0) supersymmetry whose bosonic content is one self-dual tensor and one scalar. This reduces to a vector multiplet of d = 4, N = 2 supersymmetry; 2. the multiplet of d = 6, (2, 0) supersymmetry whose bosonic content is one self-dual tensor and ﬁve scalars. This reduces to a vector multiplet of d = 4, N = 4 supersymmetry. tensor operator in CFT, a local operator whose conformal or superconformal transformation involves only the ﬁrst derivative of the transformation; synonymous with a primary ﬁeld of the conformal or superconformal symmetry. Such an operator is mapped to a highest weight state by the state–operator isomorphism.

Glossary

385

tetrad a basis of d orthonormal vector ﬁelds in d dimensions. The term tetrad originates in d = 4 (as does the equivalent term vierbein) but there is no other convenient term for general d. the theory formerly known as strings see M-theory, second deﬁnition. theory see vacuum. theta functions the family of holomorphic functions having simple periodicity properties on a torus. Thirring model the solvable quantum ﬁeld theory of a single Dirac fermion with a quartic interaction in 1 + 1 dimensions. This is equivalent under bosonization to a free scalar ﬁeld at a general real radius. threshold correction a correction to the low energy eﬀective action, and in particular to the gauge coupling, due to virtual massive particles. topological string theory a modiﬁcation of string theory without local dynamics; all observables are topological. toric geometry a generalization of the idea of projective space. Roughly speaking, this corresponds to the most general linear sigma model. toroidal compactiﬁcation the periodic identiﬁcation of one or more ﬂat dimensions. torsion a term applied to various 3-form ﬁeld strengths, so called because they appear in covariant derivatives in combination with the Christoﬀel connection. tree-level the Feynman graphs which become disconnected if one propagator is cut, or the analogous string amplitudes, the sphere and disk. These correspond to classical terms in the eﬀective action. twist 1. (verb) to deﬁne a ﬁeld in a periodic space to be aperiodic by a symmetry transformation h; 2. (noun) the aperiodicity h; 3. (verb) given a CFT or string theory, to construct a new theory using a symmetry group H. One adds closed strings twisted by any of the elements h ∈ H, and requires all states to be invariant under the transformations in H. This is equivalent to treating H as a world-sheet gauge symmetry. The term orbifold is also used as a synonym; 4. (noun, archaic) world-sheet parity. twisted state a closed string with twisted periodicity. type ... supergravity the low energy supergravity theory of the corresponding string theory. The Roman numeral signiﬁes the number of d = 10 supersymmetries, and IIA and IIB distinguish whether the two supersymmetries have opposite or identical chiralities respectively. type I superstring the theory of open and closed unoriented superstrings, which is consistent only for the gauge group SO(32). The right-movers and leftmovers, being related by the open string boundary condition, transform under the same spacetime supersymmetry.

386

Glossary

type IIA superstring a theory of closed oriented superstrings. The right-movers and left-movers transform under separate spacetime supersymmetries, which have opposite chiralities. type IIB superstring a theory of closed oriented superstrings. The right-movers and left-movers transform under separate spacetime supersymmetries, which have the same chirality. type 0 string a pair of nonsupersymmetric string theories, which have tachyons and no spacetime fermions, with the same world-sheet action as the type II theories but with diﬀerent projections on the Hilbert space. U -duality any of the dualities of a string theory, usually of a toroidally compactiﬁed type II theory. This includes the S-dualities and T-dualities, but in contrast to these includes also transformations that mix the radii and couplings. ultrashort multiplet see BPS state. unit gauge a choice of coordinate and Weyl gauges in two dimensions, such that the metric is the identity. unitary as applied to a quantum system, the property of having a conserved inner product in a positive-norm Hilbert space. unoriented string theory a string theory in which world-sheet parity-reversal Ω is a discrete gauge symmetry. The perturbation theory includes unoriented world-sheets, and the spectrum is restricted to states with Ω = +1. UV divergence a divergence arising from short distances in spacetime, usually signifying a limit to the validity of a theory. vacuum a stable Poincar´e-invariant state. The novel feature of systems with unbroken supersymmetries is the frequent appearance of degenerate but physically inequivalent vacua. While it is now clear that the diﬀerent string theories are actually diﬀerent vacua in a single theory, it is still common to use the term theory for each. Also, diﬀerent CFTs within a single string theory are sometimes referred to as diﬀerent theories rather than vacua. The term model is used to refer to string vacua whose low energy physics resembles the Standard Model. vector multiplet in d = 4, N = 1 or d = 6, N = 1 supersymmetry, the multiplet whose bosonic content is a massless vector ﬁeld. The d = 6 multiplet reduces to a vector ﬁeld plus two real scalars in d = 4, N = 2 supersymmetry. Veneziano amplitude the bosonic string tree-level amplitude for four open string tachyons. Verma module the set of states obtained by acting on a highest weight state with Virasoro raising generators in all inequivalent ways, with the requirement that all such states be linearly independent (the generators act freely). This gives a representation of the Virasoro algebra that depends on the central charge c and on the weight h of the highest weight state. This is

Glossary

387

very similar to a conformal family, except that in the case of a degenerate representation some states in the latter may vanish; the Verma module would in this case give a reducible representation. vertex operator a local operator on the string world-sheet, corresponding to a string in the initial or ﬁnal state. Virasoro algebra in a CFT, the inﬁnite-dimensional Lie algebra of the Fourier modes of the energy-momentum tensor. Virasoro–Shapiro amplitude the bosonic string tree-level amplitude for four closed string tachyons. W algebras a family of extended chiral algebras with currents of spin greater than 2. W string a proposed generalization of string theory with constraints of spin greater than 2. No complete construction exists. Ward identity a relation between the divergence of the expectation value of a conserved current and the same expectation value without the current. ˜ 0 eigenvalue, which determines the behavior weight 1. in CFT, the L0 or L of an operator under scale transformations and rotations (= conformal weight); 2. in Lie algebra, a vector of the eigenvalues of the maximal set of commuting generators of a Lie algebra, in a state of a given representation. Wess–Zumino consistency condition the condition that the second derivative of a functional integral with respect to the background ﬁelds be symmetric. This strongly constrains the form of possible anomalies in a theory. Wess–Zumino–Novikov–Witten (WZNW) model a conformally invariant nonlinear sigma model on a group manifold with an antisymmetric tensor background. Weyl anomaly an anomaly in the Weyl transformation, this determines the critical dimension in the Polyakov formalism. This is sometimes called the trace anomaly or the conformal anomaly. Weyl condition in even dimensions, the condition that a fermion have deﬁnite chirality. This deﬁnes a spinor representation of the Lorentz group, which contains half of the components of the Dirac representation of the gamma matrix algebra. Weyl transformation a position-dependent rescaling of the metric. Wilson line 1. a gauge ﬁeld with vanishing ﬁeld strength but with nontrivial parallel transport (holonomy) around nontrivial paths in spacetime. 2. the gauge-invariant operator that measures such a ﬁeld: the trace of the pathordered product of the line integral of the vector potential. Wilsonian action the action in a low energy eﬀective ﬁeld theory, which incorporates the eﬀects of higher energy virtual states. winding state a closed string whose conﬁguration is a nontrivial path in a non-simply-connected spacetime.

388

Glossary

world-sheet 1. the two-dimensional surface in spacetime swept out by the motion of a string. 2. the abstract two-dimensional parameter space used to describe the motion of a string. world-sheet duality the equivalence between Hamiltonian descriptions obtained by cutting open a string world-sheet along inequivalent circles. Important examples are the associativity condition, from the four-point sphere amplitude, and the equivalence of the open string loop and closed string tree descriptions of the cylinder. wrapped refers to a p-brane, q of whose dimensions are wound on a nontrivial compact submanifold of spacetime, leaving a (p − q)-dimensional extended object. Yang–Mills ﬁeld non-Abelian gauge ﬁeld. Zamolodchikov metric the expectation value of a pair of local operators on the sphere, giving a natural inner product for the corresponding string states. This is the metric that appears in the kinetic term in the spacetime action. zero modes 1. of a diﬀerential operator: eigenfunctions with zero eigenvalue. In a Gaussian functional integral, these would give an inﬁnite factor in the bosonic case and a zero factor in the fermionic case. In general these have a physical origin and the functional integral has an appropriate integrand to give a ﬁnite result; 2. center-of-mass modes. zero-point energy the energy due to vacuum ﬂuctuations of quantum ﬁelds. zero-slope limit the limit α → 0. Only massless string states, described by low energy ﬁeld theory, remain.

Index Page numbers in italics refer to exercises

Abelian diﬀerential 293 action (D-brane) 270–3 Born–Infeld form 271, 273 linearized 275–6 action (general) Euclidean 335 fermionic oscillator 343 harmonic oscillator 334, 339–41 in path integral 77, 331, 343 action (point particle) 9–11 Euclidean 82–3 gauge-ﬁxed 129 in curved spacetime 108 Wilson line background 264 action (spacetime) c = 1 moduli 246 closed string tachyon 194 dimensional reduction 232–5, 249, 265 higher-derivative terms 188–9, 195 Hilbert (gravitational) 83, 114, 194 inheritance principle 258 local symmetries 196 massless closed string 114–15 quadratic 275–6 open string 188–9, 196–7, 276 string ﬁeld 311–14, 327–8 tadpole terms 197–8, 225, 309 action (world-sheet) bc 50 βγ 52

Brink–Di Vecchia–Howe–Deser– Zumino 12 Euclidean 82–3 Euler term (χ) 15–16, 30, 83–4, 179, 203 Faddeev–Popov ghost 89, 157–8 gauge-ﬁxing 126–7, 131 Hilbert (gravitational) 15–16 linear dilaton 49 Liouville 324–5 massless scalar 32, 34, 346 Nambu–Goto 10–13, 15, 28, 30 and D-brane action 270 noncritical string 325–6 nonlinear sigma model 108–10, 112 T -duality 280–1 Polyakov 12–16, 28 and noncritical string 325 Euclidean 82–3 ﬂat world-sheet 32 wrapped torus 237 antisymmetric tensor background 249–50 aﬃne Lie algebra 243 annulus, see cylinder anomaly BRST 132–3, 143 from string loops 308–9 conformal same as Weyl current conservation 119, 158–9 ghost number 158–9, 177 gravitational (diﬀ) 95, 349

389

390 anomaly (cont.) Lorentz 23–4, 94 trace same as Weyl Weyl 90–7, 119, 349 from string loops 308–9 in nonlinear sigma model 111–12 in vertex operators 102–7 integrated 93–4 anticommuting c-numbers 341 antiperiodic boundary conditions in traces 212, 336, 344, 345–6 in twisted states 257–9 antisymmetric tensor ﬁeld as axion 29 background 108–15, 120 constant 249–51, 254–5 coupling to D-brane 270–2, 282 winding strings 241–2 dimensional reduction 234–5, 241 ﬁeld equation 111–12 gauge invariance 27, 109, 196 in BRST quantization 137 in light-cone quantization 27–30 in OCQ 125 spacetime action 114 vertex operator 104–7 axion 1, 6, 29 bc CFT 50–1, 75–6, 89 expectation values 176–8, 212–13 mode expansion 61–3 state–operator mapping 65–6 βγ CFT 52, 75 BRST invariance of string amplitudes 160–1, 283, 285, 287–90, 351 of vertex operators 164, 165 BRST quantization 126ﬀ, 350 and string ﬁeld theory 311–15 nilpotence 128, 130–2, 138, 143 point particle 129–31 string 131–43 anomaly 132, 143, 308–9 charge 132 current 131

Index equivalence to light-cone 137–41 equivalence to OCQ 141–3 no-ghost theorem 137–41 transformation 131 background charge 170, 204, 208 background ﬁelds 108–18 linearized, and vertex operators 108–11 Batalin–Vilkovisky (BV) formalism 128 Beltrami diﬀerential 162 Beltrami’s equation 162 Berezin integration 341 Jacobian 345 beta function (Euler) 183 beta function (ﬁeld theory) 91, 111–12, 349 string loop corrections 308–9 world-sheet corrections 112, 195 black holes 6–7, 320 Boltzmann weight 325 Borel summation 316, 352 Born–Infeld action 271, 273 bosonic string theory artifacts of 23, 219, 223–4, 246, 275, 306 reason for studying 8 bosonization 303 boundary state 226, 230 CP T symmetry 23 Calabi–Yau compactiﬁcation 352 Campbell–Baker–Hausdorﬀ formula 202 canceled propagator argument 161, 196, 288, 309 Casimir energy 22 in unitary CFT 73 mnemonic 73 central charge 48, 75–6 and BRST anomaly 132, 143 and compactiﬁcation 113, 117, 137 and density of states 213 and gravitational anomaly 95 and ground state energy 73 and Weyl anomaly 92–6

Index bc 51 βγ 52 free scalar 48 in Virasoro algebra 56 linear dilaton 49 positivity 72, 76 Chan–Paton factor 186–7 Chan–Paton states 30, 184–92 and D-branes 268–70 in cylinder amplitude 223 in M¨ obius strip 228 non-Abelian gauge interactions 188–9 unoriented string 189–92 Chern–Simons term 234–5 chiral gauge symmetry 6 closed state (BRST) 128 closed string 14 BRST quantization 136–7 light-cone quantization 25–9 OCQ 124–5 one-loop amplitudes 216–22, 226–8 poles in cylinder amplitude 225–6 230, 274–6 tree-level amplitudes 192–8 relation to open 195 vertex operators 63–8, 101–7 cocycle 239 coeﬃcient functions (OPE) 37–8 and conformal bootstrap 304 and conformal invariance 47 and matrix elements 200 and three-point amplitudes 284 determine CFT 303 generalized 303 interchange symmetry 40–1 cohomology (BRST) 128–9, 134–43, 144, 350 and amplitudes 283, 285, 289–90 Coleman–Weinberg formula 221, 229–30, 350 collective coordinates (D-brane) 269, 281–2 commutators and anticommutators canonical 17, 20, 26, 130 Jacobi identity 128, 133, 143

391

relation to OPE 54–5 wrong-sign 27–8, 72, 121 commutators and anticommutators (speciﬁc) [αµm , ανn ] 20, 26–7, 58, 62 {bn , cn } 61, 130, 133 [δτ1 , δτ2 ] 129 [GI , GJ ] (constraints) 133 [ jmi , jnj ] 243 [Lm , Ln ] 56–7, 75–6 [Lm , On ] 56 [N g , bm ] 62 [N g , cm ] 62 [pµ , xν ] 17, 20, 26, 58, 62, 130 [Q, A ] 55 [Q1 , Q2 ] 55 {QB , bm } 132 {QB , Lm } 143 {QB , QB } 132–3, 143 [ XL (z), XL (z ) ] 239 compact CFT 117, 137 compactiﬁcation 2, 6, 115–18 abstract CFT 117–18, 137 scale 110, 112–13, 116, 244–5 see also orbifold, orientifold, toroidal complex coordinates 33–4, 153 divergence theorem 33–4 complex manifold 153 conformal bootstrap 304 conformal ﬁeld theory (CFT) 43–76, 349, 351 and compactiﬁcation 115–18 c = 1 261–3, 351 compact 117, 137 coupling to curved metric 95, 97 expectation values and matrix elements 199–201 M¨ obius invariance 198–9, 205 positive genus 169–78, 201–2 zero genus 208–16 free 32–6, 49–52 matter 122, 131 mode expansions 58–63 on Riemann surface 154 periodic scalar 235–40, 242–4, 247–8

392

Index

conformal ﬁeld theory (CFT) (cont.) sewing and cutting 300–5 state–operator isomorphism 63–8 unitary 71–3, 117, 137 see also bc, βγ, free scalar, linear dilaton CFTs conformal gauge 85 and light-cone gauge 83 curvature 93 ghost action 89 OCQ 121 sphere 166 variations around 152 conformal invariance 43–5, and OPE 45–7 relation to diﬀ×Weyl invariance 85–6 conformal Killing equation 151 conformal Killing group (CKG) or vector (CKV) 149–52, 155, 157, 161 speciﬁc Riemann surfaces positive genus 167–9, 176, 178, 192, 197, zero genus 206–8, 212, 216–17 conformal transformation 44–5, 75 and Virasoro algebra 52–8 d ≥ 3 75 of energy-momentum tensor 47–9 of ghost-number current 51 conjugacy classes 260 constraints (in quantization) 28, 130, 133, 143–4, 350 contraction 39–40, 50, 65, 75, 172, 175 self- 103, 171, 338 coordinate transformation large see modular invariance spacetime 28, 109, 196, 285, 310 world-sheet see diﬀ invariance cosmological constant spacetime 222, 306, 350 world-sheet 95, 323 covariant derivative 89, 118 covariant gauge see conformal gauge critical dimension eﬀect of backgrounds 113, 120

from BRST anomaly 132, 143 from cylinder amplitude 230 from Lorentz anomaly 21–4, 27 from modular invariance 230 from OCQ 124, 143 from Veneziano amplitude 204 from Weyl anomaly 94–5 critical point 44 cross-ratio 199 cross-cap 101–2, 207–8, 227–8 state 230 current algebra 243–4 curved spacetime 108–18 cycle A- and B- 147–8, 292, 296, 304 cylinder (annulus) 78, 81 long see handle, degenerating Riemann surface 207 semi-inﬁnite 63–4 string amplitude 222–6, 229–30 calculation of D-brane tension 274–6 closed string poles 225–6, 230 divergence 224–5, 229 unoriented 226 D-brane 7, 31, 104, 268–82, 317, 351 action 270–3 gauge symmetry 268–73 half-D-brane 280 tension 273–7 Dedekind eta function 210, 216 Del Guidice–Di Vecchia–Fubini (DDF) operators 143, 240 density of states 213, 321 determinant functional 182, 338–41, 344–5 in Born–Infeld action 271 of Laplacian 170 variation 12 see also Faddeev–Popov determinant diﬀ (reparameterization) invariance 12–13 anomaly 91, 95, 349 gauge-ﬁxing 85, 129, 325

Index of vertex operators 102–3 relation to conformal invariance 44, 85–6 removes unphysical states 27–8, 117 with boundary 90 dilaton 6 and equivalence principle 115 and string coupling 112–15 background 108–15, 233–4 coupling to D-brane 270–7 open strings 196–8 orientifold plane 280 dimensional reduction 233–4, 241 eﬀect of T -duality on 248 ﬁeld equation 111–12, 246 in BRST quantization 137 in light-cone quantization 27, 29–30 in OCQ 125 linear see linear dilaton CFT moduli space 254 propagator (ﬁeld theory) 276 spacetime action 114–15, 196–8, 225, 275–6 vertex operator 104–7 dimensional reduction 235, 249 Dirichlet boundary condition 31, 266–70, 281 T -dual to Neumann 266–70 discrete gauge symmetry 260–1, 277–8 disk conformal to semi-inﬁnite cylinder 63–8 expectation values 174–5, 178 Riemann surface 168 string amplitudes 178–89, 196–8, 204–5 with D-branes 270, 274, 281–2 tadpole 197–8, 223–6 cancellation with projective plane 229 divergences analytic continuation past 181, 287 cancellation for SO(8192) 228–9 from boundary of moduli space 196, 296, 305–6

393

IR 181, 223–5, 306–10 UV 2–7, 218–20 absence in string theory 4–5, 79, 219, 223–5, 309–10 zero mode 173 doubling trick 57, 63, 66, 178 dual resonance model 348 duality see T -duality, world-sheet duality. eﬀective ﬁeld theory 110, 114, 117, 188–9, 196–7, 270–3, 309 Einstein metric 115, 194, 275 Einstein’s equation 112–13, 222 electroweak scale 222 energy-momentum tensor 13, 91 and derivative with respect to moduli 161, 210–11 as constraint 121 boundary condition 56–7 choice of 49, 95, 303 conformal transformation 47–9 trace anomaly 92–7 traceless and holomorphic 43–4 vanishes for Chan–Paton states 185 enhanced gauge symmetry 242–4, 248, 254 epsilon prescription 182, 287–8, 305–6 equivalence classes BRST 128 gauge 84, 87 OCQ 123 equivalence principle 115 Euclidean adjoint 202–3, 336 Euclidean continuation 82–3, 170, 210, 335 Euler number 96, 101, 118, 152 in string action 83–4, 112, 179, 203 exact state (BRST) 128 exceptional Lie algebras 192 expectation values 34, 36 calculations 169–78, 201–2, 208–13 exponential operator 40, 46 exterior derivative 128

394 Faddeev–Popov determinant 86–90, 154–8 on disk 176 on sphere 176 on torus 212–13, 216–17 shortcuts 179, 218 sign 156, 177, 345 Faddeev–Popov ghosts 51–2, 88–90, 126 boundary condition 90 central charge 94 vacuum 61, 141 Faddeev–Popov procedure, general 126 Feynman diagrams calculation of D-brane interaction 275–6 calculation of Weyl anomaly 119 decomposition of string amplitudes 297–306 vs string graphs 3–5, 101 ﬁducial metric 85 ﬁeld strength, three-index 109, 112, 234–5 ﬁrst-quantization 310 Fischler–Susskind mechanism 308–9 ﬁxed point orbifold 256–8 UV 4 ﬂat direction 246, 262–3, 272–3 free energy 321–2, 328 free parameters, absence of 6, 83–4, 112–13 free scalar CFT 32–41 expectation values 169–76, 208–11, 229 left-, right-moving 238 mode expansion 58–61 state–operator mapping 63–5 Fuchsian group 292–3 functional integral see path integral fundamental region 148–50 in torus amplitude 216, 219–20 gauge boson 1, 4 open string 28, 123–4, 135–6

Index amplitudes 187–9, 205 in BRST quantization 135–6 in light-cone quantization 23 in OCQ 123–4 vertex operator 107 closed string 241–3 amplitudes 242 gauge coupling 115, 234, 242–5, 276 gauge-ﬁxing dependence on gauge choice 90–1, 121–2, 126–8 diﬀ×Weyl 17–18, 25, 84–6, 131, 154–5 general 126–8 see also speciﬁc gauges gauge interaction 2, 28, 78, 184–92 gauge transformation 107 Gaussian-averaged gauges 127 general relativity 1–2, 5, 28, 108 oﬀ-shell amplitudes 104 genus 100–1, 150 large 315–17 geodesic curvature 30, 82, 96 geodesic distance 102 geodesic equation 108 ghost number 51, 62, 127 anomaly 119, 158–9, 177 of string ﬁeld 313–14 states vs operators 65–6 ghosts see Faddeev–Popov ghosts global symmetry 189 Goldstone boson 233 graded Lie algebra 143 grand uniﬁcation 2, 5 Grassmann variables 35, 88, 156–8, 341–5 gravitational coupling 114 relation to gauge coupling 234, 244–5 string coupling 194, 205 graviton coupling to D-brane 270–7 open strings 196–8 orientifold plane 280 in BRST quantization 137

Index in light-cone quantization 27, 29–30 in OCQ 125 propagator (ﬁeld theory) 276 vertex operator 104–7 gravity 1–7, 78 Green’s function 169–71, 174–6, 208, 229–30, 275 Hagedorn temperature 321–2, 352 Hamiltonian and path integral 329–32, 335–6, 343 eﬀect of backgrounds 250, 264 in partition function 209, 212–13 point particle 17, 130, 218 string 19–21, 25–6 relation to L0 54, 73, 125 handle 83, 100–1, 290–2 degenerating 97–8, 101, 224–6, 229, 295–6, 305–9 hard scattering 183–4, 194, 281–2, 317–19, 352 heterotic string 348 hidden sector 6 Higgs ﬁeld 1, 222, 245–6 high-energy scattering see hard scattering, Regge scattering high temperature 320–2 highest weight state 70–2, 122, 125, 143 and tensor (primary) operators 71 holomorphic current 42 ﬁeld 34, 56, 72 quadratic diﬀerential 152 transition functions 153–4 vector ﬁeld 152 hyperelliptic surface 294 infrared (IR) divergences 181, 223–5, 306–10 inheritance principle 258, 261 inner product bilinear (Zamolodchikov) 199–204 in sewing 301–2 relation to Hermitean 202–4 Fermi oscillator 343 Hermitean 134–5

395

positive 71–2, 116–17, 138 traceless symmetric tensor 118 instantons 316 interaction irrelevant 3 string 77–82, 104, 178ﬀ Jacobi identity 128, 133, 143 Jacobi’s abstruse identity 215 Jacobian, Berezin integral 345 Kac–Moody algebra 243 Kaluza–Klein gauge ﬁeld 2, 232–4, 241–3, 249, 257, 351 Kaluza–Klein states 232, 245 Klein bottle Riemann surface 207 string amplitude 226–9, 230, 278, 280 large order behavior 315–17 lattice even self-dual 252–3, 281 momentum 251–3, 281 Laurent expansion see mode expansion length, minimum 248, 318 level current algebra 243–4 oscillator 23, 26–7 level-matching 27, 258 light-cone coordinates 16 light-cone gauge 17 oﬀ-shell 104 light-cone quantization point particle 17 string 17–30, 349 equivalence to BRST and OCQ 137–43 linear dilaton theory 49–50, 61, 75, 204, 213, 349 as background 113, 120, 323–4, 328 Liouville ﬁeld theory 325 loop expansion 115 Lorentz invariance 12, 106–7, 116–17, 290 in light-cone gauge 20–4, 90, 94, 349

396

Index

Mandelstam variables 180 manifold complex 153 diﬀerentiable 153 Riemannian 153 mass-shell condition from vertex operators 101, 106 in BRST 131, 134, 137–8 in light-cone quantization 17, 22, 26 matrix elements, relation to path integral 199–201, 332–4 matrix models 326–7, 352 measure on moduli space 154–65 BRST invariance 160–1 derivation 154–7 gauge invariance 159–60 in terms of determinants 157–8 transition functions 161–4 see also Faddeev–Popov metric (spacetime) 9, 108–18 dimensional reduction 231–3 Einstein 115, 194, 275 internal 249ﬀ light-cone coordinates 16 sigma model 115 string 115 metric (world-line) 10 gauge-ﬁxing 129, 145–6 metric (world-sheet) constant curvature 292 ﬂat 32–3 gauge-ﬁxing 17, 84–6, 146–52 induced 11, 15 minimal area 300, 351 noncritical string 94–5, 325–6 Polyakov 12–13ﬀ mnemonics, zero-point 73–4, 125 M¨ obius group 56, 167 M¨ obius strip Riemann surface 208 string amplitude 228–9, 280 M¨ obius transformation 178–80, 192, 198–9 mode (or Laurent) expansion bc CFT 61

free scalar (X µ ) 26, 58 periodic 235–8 twisted 257 holomorphic operator 56 open strings 20, 57, 62–3 Dirichlet 267 Tab 53, 70 model, solvable 323–7 modular group 148–9, 151 and theta functions 214–16 spacetime 253–5 surfaces without 207–8, 223, 278 used to analyze asymptotics 213, 224, 227–8 modular invariance 210, 217–18, 229–30 constraints from 213, 304, 350 in orbifold compactiﬁcation 257, 259 in toroidal compactiﬁcation 237, 251–3 in twisting construction 260 needed for consistency 220 one-point function 304 moduli (spacetime) 233–4, 249, 265 moduli space 253–5 branches 262 c = 1 261–3, 351 metric on 254 moduli (world-sheet) encoded in transition functions 152–4, 161–4 for metric 147–52 moduli space 148–52, 290–4 boundary of 161, 295–6, 300; CFT asymptotics 302–3; divergences from 196, 224–6, 300, 305–9 graphical decomposition 297–300, 351 in open string ﬁeld theory 313–4, 316–7 measure on 154–65, 350 total derivative on 161, 196, 220, 288

Index vertex operator positions 150–1, 163–4 Nambu–Goto action 10–13, 15, 28, 30 Narain compactiﬁcation 251–5, 351 naturalness problem 2 Neumann boundary condition 14, 31, 62, 174 T -dual to Dirichlet 265–70 nilpotence see BRST no-ghost theorem 137–43, 350 Noether’s theorem 41–3, 74–5 anomalous 159 as local property 86 non-Abelian gauge theory 188–92, 243 from string ﬁeld theory 313 on D-brane 272–3 noncommutative geometry 272 noncritical string theory 325–7 nonlinear sigma model 108–10, 120, 349 nonperturbative eﬀects 234, 248, 269, 310, 315–7 exp(−1/gc ) 317, 327 normal ordering boundary 75, 175 conformal 36–40, 50, 60 constants 22–3, 27, 76 direct calculation 60 from state–operator mapping 72 from Virasoro algebra 59, 61 mnemonics 73, 125 creation–annihilation 60 diﬀ-invariant 102–3 relation between diﬀerent forms 59–60, 76 null state BRST 128, 136, 311 decoupling 119, 161, 196, 220, 285–95 OCQ 122–5, 143 O(k, k, R) 252–3 O(k, k, Z) 253 oﬀ-shell amplitudes 97, 103–4 old covariant quantization (OCQ) 121–5

397

equivalence to BRST and light-cone 141–3 vertex operators 125, 143, 160–1, 164, 165 one-loop amplitudes 150, 206–30 ﬁniteness 220 vs ﬁeld theory 218–20 open string theory 14 BRST quantization 131–6 Dirichlet 266–70 light-cone quantization 17–25 must include closed strings 81–2, 225 OCQ 122–5 one-loop amplitudes 222–6, 228–9 tree–level amplitudes 178–92 relation to closed 195 vertex operators 66, 107 operator calculations 201–2 operator equation 35–6, 333–4 operator product expansion (OPE) 36–41, 74–6 and conformal invariance 45–9 and Ward identity 42–3 associativity 70, 201, 304 convergence 37–8 derivation CFT 68–70 free ﬁeld theory 37–8 generalized 302–3 interchange symmetry 40–1 Laurent expansion 70–1 nonzero separation 48 relation to commutator 54–7 z-, ¯z -dependence 47 operator product expansions (speciﬁc) Ai Aj 47 b b, b c, c c 50 β β, β γ, γ γ 52 DDF operators 240 ∂X µ ∂Xµ 211 : eik·X : : eik ·X : 40, 251 : F : : G : 39 jB b, jB c 132 jB Om 132 jB jB 132

398

Index

operator product expansions (speciﬁc) (cont.) jB V 165 j i j j 243 j µ : eik·X : 43 Oi Oj 200 T A 45–6 T eik·X 46 T j 51 T jB 133 T O 46 T T 47–8 ˜ 47–8 TT V V 284, 289, 327 XL XL , XL XR , XR XR 238 X µ X ν 38 orbifold 256–63, 351 ﬁxed points 256–8 inheritance principle 258, 261 modular invariance 259–60 twisted states 31, 256–61 vertex operators 258, 351 twisting construction 259–63 oriented string theory 29 see complement of unoriented string theory orientifold 277–80 ﬁxed planes 278–80 local physics oriented 278 outline of both volumes 8–9 P SL(2, C) 168, 192, 197–9, 205 P SL(2, R) 168, 178, 199, 205, 255, 292–3 P SL(2, Z) 148, 255 parity spacetime 6, 255, 277 world-sheet 29, 200, 255, 345 action on Chan–Paton factors 189–92 gauging 277 in one-loop amplitudes 207–8, 226–8 partition function 209 bc 212 general CFT 213

orbifold 258–61 scalar CFT 209–11 thermal 320–2, 328 toroidal compactiﬁcation 237, 250–2, 281 path (functional) integrals 32–6 and expectation values 34 canonical commutators from 54–5, 58–61 Euclidean rotation 82–3, 170, 210, 335 evaluation by integrating Weyl anomaly 93–4 by operator methods 201–2, 209–10, 212 direct 157–9, 169–72, 176–7, 337–41, 344–5 using holomorphicity 172–4, 177, 210–11 existence 332 fermionic 341–5 for traces 335–6, 344 Hamiltonian 331 introduction 329–46 Lagrangian 331 of total derivative 34, 333, 341 operator equations in 35, 333 Polyakov 82–4 gauge-ﬁxing 84–90, 154–9 in curved spacetime 108–12 Weyl anomaly in 90–7 relation to operator expressions via cutting 55, 199–201, 300–5, 332–4 representation of string amplitudes 77–82, 97–101 twisted 260, 336 used to derive state–operator mapping 66–8 Pauli–Villars regulator 91, 340 period matrix 293 periodic boundary condition 14, 31, 52 perturbation theory, string 5, 101, 283–310 physical state BRST 127–9

Index b0 condition 131, 134, 136, 289–90 vertex operators 164 OCQ 121–3 L0 condition 125 vertex operators 125, 143, 160–1, 164 Planck length 3, 116 Planck mass 3, 245 plumbing ﬁxture 294–305 Poincar´e invariance 10, 12–16, 91, 185 and tadpoles 225 four-dimensional 16 of boundary conditions 14–15 point particle action 9–11 and Wilson line 264 BRST quantization 129–31 in curved spacetime 108 light-cone quantization 17 path integral 77, 82–3, 164–5 vacuum loop 145–6, 164, 218, 220–2 Poisson resummation formula 237, 250, 252 Polyakov action see action, Polyakov Polyakov path integral see path integral, Polyakov potential energy c = 1 moduli 246 from dilaton tadpoles 197–8, 225, 309 from quartic Yang–Mills interaction 265, 272 primary ﬁeld see tensor operator projective plane expectation values 176, 178 Riemann surface 168–9 string amplitudes 196–8 tadpole 197–8, 227–8 cancellation with disk 229 propagator dilaton 275–6 graviton 275–6 photon 290 point particle 115, 164–5 pole 225–6, 307–8

399

string 297, 305–6, 313–14 proton decay 1 puncture 296–300 QCD string 25 quantization canonical 17, 19–20, 26, 130, 249–50, 164 path integral 32–6, 77–82, 329–46 see also BRST, light-cone, OCQ, path integrals quantum ﬁeld theory 1–7, 104, 113, 310, 331–2 UV divergence 218–20 world–sheet 13–15, 32ﬀ, 108–10, 118 quantum gravity 2–7, 25 quantum mechanics 1, 116, 329–46 Regge behavior 183–4, 193, 320, 352 Regge slope 11, 24, 112 renormalization 38, 102–3, 189, 332, 340 renormalization group equation 4, 91, 111 Ricci curvature spacetime 111–12, 114 dimensional reduction 232–3 world-sheet 15, 85, 92–3 Riemann–Roch theorem 152, 158–9, 293 Riemann surface 100–1, 152–4, 161–4, 351 sewing 294–300, 351 speciﬁc 166–9, 206–8, 290–4 Riemann tensor 85 SL(2, C) (algebra) 71 invariant state 71, 203 SL(2, R) (algebra) 56, 71 SO(D − 2) 23–4, 27, 30 SO(D − 1) 23–4, 27, 30–1 SO(n) 30, 191–2, 228 on D-brane 279–80 SO(8192) 228–9, 280 Sp(k) 30, 191–2, 228, 279 SU(n) 189

400

Index

SU(2) × SU(2) radius 242–8, 261–3 on D-brane 264–5, 269–72, 279 SU(3) × SU(2) × U(1) model 1 see also Standard Model S-matrix 97–101, 103–4 calculations 178–205 expression for 99, 155–7, 164 unitarity 181–2, 186–7, 193–4, 196, 285–90, 306 vs Green’s function 104 s, t, u (Mandelstam variables) 180 saturating geodesic 300 scale transformation 44–8, 90–1 Schottky group 290–1, 351 Schwarzian 48 Schwinger–Dyson equation 334 self-contractions 103, 171, 338 self-dual radius 242–8, 261–3 sewing 332, 351 circles 298–9 construction of general surface from three-punctured spheres 296–7 of CFTs 300–5 and conformal bootstrap 303–5 of Riemann surfaces 294–300 parameter 294–5 used to analyze string amplitudes 305–7 Siegel gauge 313 sigma model metric 115 soliton 104, 235 sphere expectation values 174–8, 198–204 Riemann surface 166–8 string amplitudes 192–6, 204–5, 283–90 thrice-punctured 296–7, 303 cancellation with projective plane 229 spin 23–4, 184, 204 in CFT 46, 213 in Coleman–Weinberg formula 221 spurious state 122 Standard Model 1–7, 25, 112, 310, 348–9

state–operator isomorphism 63–73, 101–2 explicit form 65 path integral derivation 66–8 vertex operators 101–2, 164, 199–201 Stirling’s approximation 184, 315 string coupling 83–4 and dilaton 112–13 as overall factor in action 115 in vertex operators 101–2, 107, 179 position-dependent 323–4 relation between open and closed 83–4, 276 relation to gauge coupling 187, 234 relation to gravitational coupling 194, 205 string ﬁeld theory 310–17, 352 string length scale 25, 110, 244, 248 string metric 115 string tension 11, 30 subtraction 39, 47, 60 supersymmetry 2, 6–7, 222, 283, 315, 348 supersymmetry breaking 222, 233–4 T -duality 247–8, 280–1 and D-branes 268–73 as gauge symmetry 248 O(k, k, Z) 253–5 open string theory 265–8 reverses winding and momentum 247, 268 unoriented string theory 277–80 tachyon amplitudes 178–84, 187, 192–5, 204–5, 217, 229–30 artifact of bosonic string theory 23, 219, 223–4, 246, 275, 306 in BRST quantization 135 in light-cone quantization 23, 27, 29–30 in linear dilaton background 323–4, 328 in OCQ 123

Index present in all bosonic string theories 117 vertex operator 101, 107 tadpole 197, 309 cancellation for SO(8192) 229 disk 197, 223–6, 228–9 projective plane 197, 227–9 torus 307 target space 109 Teichm¨ uller parameters 148 temperature 320–2, 328 tensor operator (primary ﬁeld) 46–7, 76 expectation values 199 maps to highest weight state 71 OPEs 47, 200, 303 tetrad spacetime 250–1 world-line 10, 129, 145–6 world-sheet 83 theta functions 214–16, 229, 350 in expectation values 208–9, 211, 228, 259 toroidal compactiﬁcation 231–55, 236–80, 280–1, 351 abstract (Narain) description 251–4 in CFT 235–40 in ﬁeld theory 231–5 in string theory 241–8 of several dimensions 249–55 of two dimensions 255 open string 263–8 torus and partition function 209 expectation values 208–13 one-point function 304 in twisted CFT 260 Riemann surface 146–50, 154, 206–7 string amplitude 216–22, 229–30, 237, 250 ﬁniteness 218–20 trace of energy-momentum tensor 13, 43, 92–7, 111 partition function 209, 335–6, 344, 345–6

401

transition functions 153–4, 161–4 unoriented 277 transverse Hilbert space 138 oscillators 20–1, 26, 125, 217–18 tree-level amplitudes 178–98, 283–90, 350–1 inheritance principle 258, 261 open–closed relation 195, 350 poles 181–2, 193–4, 196, 287–8 twisted ﬁelds 31, 257, 259, 336, 345–6 twisted state 31, 257–61 twisting (orbifold) construction 259–63 U(n) 30, 185, 188–9, 230 ultraviolet (UV) divergences see under divergences unit gauge 85 unitarity see under S-matrix unitary CFT 71–3, 117, 137, 213 unoriented string theory 29–30, 80–1 Chan–Paton states 189–92 one-loop amplitudes 226–9 orientation-reversal as gauge symmetry 149, 153, 277–9 Riemann surfaces 80–1, 100–1 T -duality 277–80, 351 vacuum amplitude ﬁeld theory 218 physics of 220–2 string 198, 216–30, 274–6 vacuum degeneracy 7, 118, 233–4, 261, 348 Veneziano amplitude 178–84, 204, 348 vertex operators 63–8, 74, 97–102, 349 general construction 63–6 must be on mass-shell 101–3, 113 normalization 101–2, 107, 187, 194, 276, 284 OCQ 125, 143, 160–1, 164, 165 Polyakov approach 102–7, 119 positions 99–100 coincident limit 195–6, 306–9 in speciﬁc amplitudes 178–80, 192–3, 204, 216–17

402 vertex operators (cont.) positions (cont.) relation to moduli and CKVs 149–52, 154–7, 163–4 speciﬁc 101–2, 107 twisted state 258, 351 winding state 238–9, 243–5, 251 Virasoro algebra 52–7, 75–6 Virasoro generators as constraints 121–2, 133 centered on operators 70 L−1 · as derivative 71 L0 · as weight 71 Hermiticity 71 oscillator expressions 59, 61–2 winding state 238 Virasoro–Shapiro amplitude 192–5, 204–5 Ward identity 41–3, 55 as local property 86 for conformal symmetry 45–6 weight (conformal) 46–7, 71–3 of vertex operators 101, 125, 143 positivity 72 Wess–Zumino consistency condition 96 Weyl anomaly see under anomaly Weyl transformation 13, 15–16, 30 as redundancy 13, 84, 117, 326 gauge-ﬁxing of 17–18, 85 ghosts trivial 88, 131

Index of curvature 15, 85, 114 of nonlinear sigma model 110–12 of vertex operators 102–7 relation to conformal transformation 86 spacetime 114, 233, 254 Wick’s theorem 60 Wilson line 263–8, 279–80 breaks gauge symmetry 265, 270 winding state 31, 235–9 Dirichlet open string 266–7 massless as R → 0 247 vertex operators 238–9, 243–5, 251 world-sheet 13–14 world-sheet duality 287, 332 cylinder 226, 274 wrapped D-brane 273 world-sheet 237, 250 Yukawa couplings 1, 79 Zamolodchikov inner product 200 zero modes ghost 157–9, 161, 176, 212 X µ 169–70, 173 zero-norm states see null states zero-point energy 59–61, 72 heuristic 21–2, 26 mnemonics 73, 125 zero-slope limit 188

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